# Foundations

## Properties of Real Numbers

### Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add  or  the results are the same—they both equal 17. So,  The order in which we add does not matter!
Similarly, when multiplying two numbers, the order does not affect the result. If we multiply  or  the results are the same—they both equal 72. So,  The order in which we multiply does not matter!
These examples illustrate the Commutative Property.
The Commutative Property has to do with order. We subtract  and , and see that  Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.
Division is not commutative either. Since  changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!
  Addition and multiplication are commutative.
  Subtraction and division are not commutative.

When adding three numbers, changing the grouping of the numbers gives the same result. For example,  since each side of the equation equals 17.
This is true for multiplication, too. For example,  since each side of the equation equals 5.
These examples illustrate the Associative Property.
The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.
When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.

### Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity. The Identity Property of Addition that states that for any real number  and
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity. The Identity Property of Multiplication that states that for any real number  and
We summarize the Identity Properties here.
What number added to 5 gives the additive identity, 0? We know
The missing number was the opposite of the number!
We call  the additive inverse of  The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number
What number multiplied by  gives the multiplicative identity, 1? In other words,  times what results in 1? We know
The missing number was the reciprocal of the number!
We call  the multiplicative inverse of a. The reciprocal of a number is its multiplicative inverse. This leads to the Inverse Property of Multiplication that states that for any real number
We’ll formally state the inverse properties here.
The Identity Property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.
What about division involving zero? What is  Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,
We can check division with the related multiplication fact. So we know  because
Now think about dividing by zero. What is the result of dividing 4 by  Think about the related multiplication fact:
Is there a number that multiplied by 0 gives  Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4. We conclude that there is no answer to  and so we say that division by 0 is undefined.
We summarize the properties of zero here.
We will now practice using the properties of identities, inverses, and zero to simplify expressions.
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.
The next example makes us aware of the distinction between dividing 0 by some number or some number being divided by 0.

### Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?
You can think about the dollars separately from the quarters. They need 3 times $9 so $27 and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.
Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in  would look like this:
Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.
When we distribute a negative number, we need to be extra careful to get the signs correct!
In the next example, we will show how to use the Distributive Property to find the opposite of an expression.
There will be times when we’ll need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the Distributive Property, which removes the parentheses. The next two examples will illustrate this.
All the properties of real numbers we have used in this chapter are summarized here.

### Key Concepts

### Section Exercises

### Practice Makes Perfect

Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter Review Exercises

### Use the Language of Algebra

Identify Multiples and Factors
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.

### Integers

Simplify Expressions with Absolute Value
In the following exercise, fill in  or  for each of the following pairs of numbers.
In the following exercises, simplify.
Add and Subtract Integers
In the following exercises, simplify each expression.
Multiply and Divide Integers
In the following exercise, multiply or divide.
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
For the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercise, solve.

### Fractions

Simplify Fractions
In the following exercises, simplify.
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
Add and Subtract Fractions
In the following exercises, perform the indicated operation.
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.

### Decimals

Round Decimals
Add and Subtract Decimals
In the following exercises, perform the indicated operation.
Multiply and Divide Decimals
In the following exercises, perform the indicated operation.
Convert Decimals, Fractions and Percents
In the following exercises, convert each decimal to a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each decimal to a percent.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.

### Properties of Real Numbers

Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Practice Test

In the following exercises, simplify each expression.


# Solving Linear Equations

## Introduction

Imagine being a pilot, but not just any pilot—a drone pilot. Drones, or unmanned aerial vehicles, are devices that can be flown remotely. They contain sensors that can relay information to a command center where the pilot is located. Larger drones can also carry cargo. In the near future, several companies hope to use drones to deliver materials and piloting a drone will become an important career. Law enforcement and the military are using drones rather than send personnel into dangerous situations. Building and piloting a drone requires the ability to program a set of actions, including taking off, turning, and landing. This, in turn, requires the use of linear equations. In this chapter, you will explore linear equations, develop a strategy for solving them, and relate them to real-world situations.


# Solving Linear Equations

## Use a General Strategy to Solve Linear Equations

### Solve Linear Equations Using a General Strategy

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!
To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation.
There are many types of equations that we will learn to solve. In this section we will focus on a linear equation.
To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.
These steps are summarized in the General Strategy for Solving Linear Equations below.
We can solve equations by getting all the variable terms to either side of the equal sign. By collecting the variable terms on the side where the coefficient of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives.

### Classify Equations

Whether or not an equation is true depends on the value of the variable. The equation  is true when we replace the variable, x, with the value  but not true when we replace x with any other value. An equation like this is called a conditional equation. All the equations we have solved so far are conditional equations.
Now let’s consider the equation  Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
Solve:
This means that the equation  is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable is called an identity.
What happens when we solve the equation
Solve:
Solving the equation  led to the false statement  The equation  will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.
The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction.
We summarize the methods for classifying equations in the table.

### Solve Equations with Fraction or Decimal Coefficients

We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.
We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator (LCD) of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called clearing the equation of fractions.
To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators.
Notice in the previous example, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve. We then used the General Strategy for Solving Linear Equations.
In the next example, we’ll distribute before we clear the fractions.
When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD—even if it does not contain a fraction.
Some equations have decimals in them. This kind of equation may occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example,  and  So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.
The next example uses an equation that is typical of the ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form.

### Key Concepts

1. How to determine whether a number is a solution to an equation

2. How to Solve Linear Equations Using a General Strategy

3. How to Solve Equations with Fraction or Decimal Coefficients

### Practice Makes Perfect

Solve Equations Using the General Strategy
In the following exercises, determine whether the given values are solutions to the equation.
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
Solve Equations with Fraction or Decimal Coefficients
In the following exercises, solve each equation with fraction coefficients.
In the following exercises, solve each equation with decimal coefficients.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Solving Linear Equations

## Use a Problem Solving Strategy

Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.
Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.
  I think I can! I think I can!
  While word problems were hard in the past, I think I can try them now.
  I am better prepared now—I think I will begin to understand word problems.
  I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that  with word problems.
  It may take time, but I can begin to solve word problems.
You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

### Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.
We summarize an effective strategy for problem solving.

### Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. Therefore, if we define the first integer as n, the next consecutive integer is  The one after that is one more than  so it is  which is
We will use this notation to represent consecutive integers in the next example.
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
Notice each integer is two more than the number preceding it. If we call the first one n, then the next one is  The one after that would be  or
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.
Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.
When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.

### Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.
Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.
Remember to put the answer in the form requested. In the next example we are looking for the percent.
It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change.
To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.
Applications of discount and mark-up are very common in retail settings.
When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.
The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

### Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.
The amount of money you initially deposit into a bank is called the principal, P, and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.
In either case, the interest is computed as a certain percent of the principal, called the rate of interest, r. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is saved or borrowed.
Interest is calculated as simple interest or compound interest. Here we will use simple interest.
The formula we use to calculate interest is  To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.
There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.
In the next example, we are asked to find the principal—the amount borrowed.

### Key Concepts

1. How To Use a Problem Solving Strategy for Word Problems

2. How To Find Percent Change

3. Discount
4. Mark-up
5. Simple Interest
If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:

### Practice Makes Perfect

Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Word Problems
In the following exercises, solve each number word problem.
Solve Percent Applications
In the following exercises, translate and solve.
In the following exercises, solve.
In the following exercises, solve.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate (Round to the nearest tenth of a percent if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.
Solve Simple Interest Applications
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Solving Linear Equations

## Solve a Formula for a Specific Variable

### Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.
It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.
Geometric formulas often need to be solved for another variable, too. The formula  is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.
In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.
The next example uses the formula for the surface area of a right cylinder.
Sometimes we might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

### Use Formulas to Solve Geometry Applications

In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.
When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.
The next example involves the area of a triangle. The area of a triangle is one-half the base times the height. We can write this as  where b = length of the base and h = height.
In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is  We can write this as a formula:  Also, since the triangle is a right triangle, we remember that a right triangle has one  angle.
Here, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.
The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.
We will use the Pythagorean Theorem in the next example.
The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as  where L is the length and  is the width. To solve the example, we will need to define the length in terms of the width.
The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as  where a, b, and c are the lengths of the sides.
Applications of these geometric properties can be found in many everyday situations as shown in the next example.

### Key Concepts

1. How To Solve Geometry Applications

2. The Pythagorean Theorem

### Practice Makes Perfect

Solve a Formula for a Specific Variable
In the following exercises, solve the given formula for the specified variable.
In the following exercises, solve for the formula for y.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.
In the following exercises, solve using a geometry formula.
In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Solving Linear Equations

## Solve Mixture and Uniform Motion Applications

### Solve Coin Word Problems

Using algebra to find the number of nickels and pennies in a piggy bank may seem silly. You may wonder why we just don’t open the bank and count them. But this type of problem introduces us to some techniques that will be useful as we move forward in our study of mathematics.
If we have a pile of dimes, how would we determine its value? If we count the number of dimes, we’ll know how many we have—the number of dimes. But this does not tell us the value of all the dimes. Say we counted 23 dimes, how much are they worth? Each dime is worth $0.10—that is the value of one dime. To find the total value of the pile of 23 dimes, multiply 23 by $0.10 to get $2.30.
The number of dimes times the value of each dime equals the total value of the dimes.
This method leads to the following model.
If we had several types of coins, we could continue this process for each type of coin, and then we would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.
The steps for solving a coin word problem are summarized below.

### Solve Ticket and Stamp Word Problems

Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.
In most of our examples so far, we have been told that one quantity is four more than twice the other, or something similar. In our next example, we have to relate the quantities in a different way.
Suppose Aniket sold a total of 100 tickets. Each ticket was either an adult ticket or a child ticket. If he sold 20 child tickets, how many adult tickets did he sell?
  Did you say “80”? How did you figure that out? Did you subtract 20 from 100?
If he sold 45 child tickets, how many adult tickets did he sell?
  Did you say “55”? How did you find it? By subtracting 45 from 100?
Now, suppose Aniket sold x child tickets. Then how many adult tickets did he sell? To find out, we would follow the same logic we used above. In each case, we subtracted the number of child tickets from 100 to get the number of adult tickets. We now do the same with x.
We have summarized this in the table.
We will apply this technique in the next example.

### Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. In mixture problems, we are often mixing two quantities, such as raisins and nuts, to create a mixture, such as trail mix. In our tables we will have a row for each item to be mixed as well as one for the final mixture.

### Solve Uniform Motion Applications

When you are driving down the interstate using your cruise control, the speed of your car stays the same—it is uniform. We call a problem in which the speed of an object is constant a uniform motion application. We will use the distance, rate, and time formula,  to compare two scenarios, such as two vehicles travelling at different rates or in opposite directions.
Our problem solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the coin, ticket, and stamp applications.
The steps are listed here for easy reference:
In , we had two bikers traveling the same distance. In the next example, two people drive toward each other until they meet.
As you read the next example, think about the relationship of the distances traveled. Which of the previous two examples is more similar to this situation?
It is important to make sure that the units match when we use the distance rate and time formula. For instance, if the rate is in miles per hour, then the time must be in hours.
In the distance, rate and time formula, time represents the actual amount of elapsed time (in hours, minutes, etc.). If a problem gives us starting and ending times as clock times, we must find the elapsed time in order to use the formula.

### Key Concepts

1. Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the model

2. How to solve coin word problems.

3. How To Solve a Uniform Motion Application

### Practice Makes Perfect

Solve Coin Word Problems
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Solve Mixture Word Problems
In the following exercises, solve each mixture word problem.
Solve Uniform Motion Applications
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Solving Linear Equations

## Solve Linear Inequalities

### Graph Inequalities on the Number Line

What number would make the inequality  true? Are you thinking, “x could be four”? That’s correct, but x could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality
We show all the solutions to the inequality  on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.
We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express  as  The symbol  is read as “infinity.” It is not an actual number.
 shows both the number line and the interval notation.
We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.
The inequality  means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at  We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See .
There is no lower end to those numbers. We write  in interval notation as  The symbol  is read as “negative infinity.”  shows both the number line and interval notation.
The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.
What numbers are greater than two but less than five? Are you thinking say, ? We can represent all the numbers between two and five with the inequality  We can show  on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See .

### Solve Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms,  or
When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:
Notice that the inequality sign stayed the same.
This leads us to the Addition and Subtraction Properties of Inequality.
What happens to an inequality when we divide or multiply both sides by a constant?
Let’s first multiply and divide both sides by a positive number.
The inequality signs stayed the same.
Does the inequality stay the same when we divide or multiply by a negative number?
Notice that when we filled in the inequality signs, the inequality signs reversed their direction.
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
This gives us the Multiplication and Division Property of Inequality.
When we divide or multiply an inequality by a:
1. positive number, the inequality stays the same.
2. negative number, the inequality reverses.
Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”
Be careful when you multiply or divide by a negative number—remember to reverse the inequality sign.
Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.
When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don’t have to multiply or divide by a negative—which means we don’t have to remember to reverse the inequality sign.
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.
We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.

### Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious.  shows some common phrases that indicate inequalities.

### Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.
We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.
Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.
There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.

### Key Concepts

1. Inequalities, Number Lines, and Interval Notation

2. Linear Inequality

3. Addition and Subtraction Property of Inequality

4. Multiplication and Division Property of Inequality

5. Phrases that indicate inequalities

### Practice Makes Perfect

Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Linear Inequalities
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate to an Inequality and Solve
In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.
Solve Applications with Linear Inequalities
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Solving Linear Equations

## Solve Compound Inequalities

### Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.
To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.
To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.
Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.
To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.
For the compound inequality  and  we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See .
We can see that the numbers between  and  are shaded on both of the first two graphs. They will then be shaded on the solution graph.
The number  is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.
The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.
This is how we will show our solution in the next examples.
Sometimes we have a compound inequality that can be written more concisely. For example,  and  can be written simply as  and then we call it a double inequality. The two forms are equivalent.
To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.
When written as a double inequality,  it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.
Another way to graph the solution of  is to graph both the solution of  and the solution of  We would then find the numbers that make both inequalities true as we did in previous examples.

### Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.
Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.
To write the solution in interval notation, we will often use the union symbol,  to show the union of the solutions shown in the graphs.

### Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.
Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality. Last, we will solve the compound inequality.

### Key Concepts

1. How to solve a compound inequality with “and”

2. Double Inequality

3. How to solve a compound inequality with “or”

### Practice Makes Perfect

Solve Compound Inequalities with “and”
In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Applications with Compound Inequalities
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Solving Linear Equations

## Solve Absolute Value Inequalities

### Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value.
We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:
 is 5 units away from 0, so
 is 5 units away from 0, so
 illustrates this idea.
For the equation  we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and  are five units from zero on the number line. They are the solutions to the equation.
The solution can be simplified to a single statement by writing  This is read, “x is equal to positive or negative 5”.
We can generalize this to the following property for absolute value equations.
To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
The steps for solving an absolute value equation are summarized here.
Remember, an absolute value is always positive!
Some of our absolute value equations could be of the form  where u and v are algebraic expressions. For example,
How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if  then  or
This tells us that
This leads us to the following property for equations with two absolute values.
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

### Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.
Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation  we saw that both 5 and  are five units from zero on the number line. They are the solutions to the equation.
What about the inequality  Where are the numbers whose distance is less than or equal to 5? We know  and 5 are both five units from zero. All the numbers between  and 5 are less than five units from zero. See .
In a more general way, we can see that if  then  See .
This result is summarized here.
After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.

### Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.
We started with the inequality  We saw that the numbers whose distance is less than or equal to five from zero on the number line were  and 5 and all the numbers between  and 5. See .
Now we want to look at the inequality  Where are the numbers whose distance from zero is greater than or equal to five?
Again both  and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than  and greater than 5 on the number line. See .
In a more general way, we can see that if  then  or  See .
This result is summarized here.

### Solve Applications with Absolute Value

Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.

### Key Concepts

1. Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as  and  for all numbers.
Absolute values are always greater than or equal to zero.
2. Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
3. How to Solve Absolute Value Equations

4. Equations with Two Absolute Values
For any algebraic expressions, u and v,
5. Absolute Value Inequalities with  or
For any algebraic expression, u, and any positive real number, a,
6. How To Solve Absolute Value Inequalities with  or

7. Absolute Value Inequalities with  or
For any algebraic expression, u, and any positive real number, a,
8. How To Solve Absolute Value Inequalities with  or

### Section Exercises

### Practice Makes Perfect

Solve Absolute Value Equations
In the following exercises, solve.
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter Review Exercises

### Use a General Strategy to Solve Linear Equations

Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, determine whether each number is a solution to the equation.
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
Solve Equations with Fraction or Decimal Coefficients
In the following exercises, solve each equation.

### Use a Problem-Solving Strategy

Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems.
Solve Number Word Problems
In the following exercises, solve each number word problem.
Solve Percent Applications
In the following exercises, translate and solve.
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.

### Solve a formula for a Specific Variable

Solve a Formula for a Specific Variable
In the following exercises, solve the formula for the specified variable.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.

### Solve Mixture and Uniform Motion Applications

Solve Coin Word Problems
In the following exercises, solve.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Solve Mixture Word Problems
In the following exercises, solve.
Solve Uniform Motion Applications
In the following exercises, solve.

### Solve Linear Inequalities

Graph Inequalities on the Number Line
In the following exercises, graph the inequality on the number line and write in interval notation.
Solve Linear Inequalities
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate Words to an Inequality and Solve
In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.
Solve Applications with Linear Inequalities
In the following exercises, solve.

### Solve Compound Inequalities

Solve Compound Inequalities with “and”
In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Applications with Compound Inequalities
In the following exercises, solve.

### Solve Absolute Value Inequalities

Solve Absolute Value Equations
In the following exercises, solve.
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve. Graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.

### Practice Test

In the following exercises, solve each equation.
In the following exercises, graph the inequality on the number line and write in interval notation.
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
In the following exercises, translate to an equation or inequality and solve.


# Graphs and Functions

## Introduction

Imagine visiting a faraway city or even outer space from the comfort of your living room. It could be possible using virtual reality. This technology creates realistic images that make you feel as if you are truly immersed in the scene and even enable you to interact with them. It is being developed for fun applications, such as video games, but also for architects to plan buildings, car companies to design prototypes, the military to train, and medical students to learn.
Developing virtual reality devices requires modeling the environment using graphs and mathematical relationships. In this chapter, you will graph different relationships and learn ways to describe and analyze graphs.


# Graphs and Functions

## Graph  Linear Equations in Two Variables

### Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the “coordinate plane.”
The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that the order is important.
What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is  The point  has a special name. It is called the origin.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point  as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through  Then, locate 3 on the y-axis and sketch a horizontal line through  Now, find the point where these two lines meet—that is the point with coordinates  See .
Notice that the vertical line through  and the horizontal line through  are not part of the graph. We just used them to help us locate the point
When one of the coordinate is zero, the point lies on one of the axes. In  the point  is on the y-axis and the point  is on the x-axis.
The signs of the x-coordinate and y-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:
Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form  An equation of this form is called a linear equation in two variables.
Here is an example of a linear equation in two variables, x and y.
The equation  is also a linear equation. But it does not appear to be in the form  We can use the Addition Property of Equality and rewrite it in  form.
By rewriting  as  we can easily see that it is a linear equation in two variables because it is of the form  When an equation is in the form  we say it is in standard form of a linear equation.
Most people prefer to have A, B, and C be integers and  when writing a linear equation in standard form, although it is not strictly necessary.
Linear equations have infinitely many solutions. For every number that is substituted for x there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair  When we substitute these values of x and y into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.
Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.
A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!

### Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.
The steps to take when graphing a linear equation by plotting points are summarized here.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.
If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for x. But the arithmetic is easier if we make “good” choices for the values of x. This way we will avoid fractional answers, which are hard to graph precisely.

### Graph Vertical and Horizontal Lines

Some linear equations have only one variable. They may have just x and no y, or just y without an x. This changes how we make a table of values to get the points to plot.
Let’s consider the equation  This equation has only one variable, x. The equation says that x is always equal to so its value does not depend on y. No matter what is the value of y, the value of x is always
So to make a table of values, write  in for all the x-values. Then choose any values for y. Since x does not depend on y, you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .
Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line.
What if the equation has y but no x? Let’s graph the equation  This time the y-value is a constant, so in this equation, y does not depend on x. Fill in 4 for all the y’s in  and then choose any values for x. We’ll use 0, 2, and 4 for the x-coordinates.
In this figure, we have graphed a horizontal line passing through the y-axis at 4.
What is the difference between the equations  and
The equation  has both x and y. The value of y depends on the value of x, so the y -coordinate changes according to the value of x. The equation  has only one variable. The value of y is constant, it does not depend on the value of x, so the y-coordinate is always 4.
Notice, in the graph, the equation  gives a slanted line, while  gives a horizontal line.

### Find x- and y-intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x-axis and the y-axis. These points are called the intercepts of a line.
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the x-axis. See .
Now, let’s look at the points where these lines cross the y-axis.
Do you see a pattern?
For each line, the y-coordinate of the point where the line crosses the x-axis is zero. The point where the line crosses the x-axis has the form  and is called the x-intercept of the line. The x-intercept occurs when y is zero.
In each line, the x-coordinate of the point where the line crosses the y-axis is zero. The point where the line crosses the y-axis has the form  and is called the y-intercept of the line. The y-intercept occurs when x is zero.
Recognizing that the x-intercept occurs when y is zero and that the y-intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x-intercept, let  and solve for x. To find the y-intercept, let  and solve for y.

### Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
The steps to graph a linear equation using the intercepts are summarized here.
When the line passes through the origin, the x-intercept and the y-intercept are the same point.

### Key Concepts

1. Points on the Axes

2. Quadrant

3. Graph of a Linear Equation: The graph of a linear equation  is a straight line.
Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line.
4. How to graph a linear equation by plotting points.

5.

6. Find the

7. How to graph a linear equation using the intercepts.

### Practice Makes Perfect

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
In the following exercises, for each ordered pair, decide
ⓐ is the ordered pair a solution to the equation? ⓑ is the point on the line?
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
Find
In the following exercises, find the x- and y-intercepts on each graph.
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Mixed Practice
In the following exercises, graph each equation.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
Confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
With some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
No, I don’t get it. This is a warning sign and you must address it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Graphs and Functions

## Slope of a Line

### Find the Slope of a Line

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter.
In mathematics, the measure of the steepness of a line is called the slope of the line.
The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. and as you ski or jog down a hill, you definitely experience slope.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope is a rate of change. See .
To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run,
How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line,  we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.

Let’s also consider a vertical line, the line  as shown in the graph.

The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line  is undefined.
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.
Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.
We have seen that an ordered pair  gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol  be used to represent two different points? Mathematicians use subscripts to distinguish the points.
We will use  to identify the first point and  to identify the second point.
If we had more than two points, we could use  and so on.
Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points  and  as shown in this graph.

We’ve shown that  is really another version of  We can use this formula to find the slope of a line when we have two points on the line.
The slope is:

### Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
We can also graph a line when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

### Graph a Line Using its Slope and Intercept

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.
See . Let’s look at the graph of the equation  and find its slope and y-intercept.
The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:
The y-intercept is
Look at the equation of this line.
Look at the slope and y-intercept.
When a linear equation is solved for y, the coefficient of the x term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation  is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”
Let’s practice finding the values of the slope and y-intercept from the equation of a line.
We have graphed a line using the slope and a point. Now that we know how to find the slope and y-intercept of a line from its equation, we can use the y-intercept as the point, and then count out the slope from there.
Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines.

### Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?
While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier.
Generally, plotting points is not the most efficient way to graph a line. Let’s look for some patterns to help determine the most convenient method to graph a line.
Here are five equations we graphed in this chapter, and the method we used to graph each of them.
Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.
In equations #3 and #4, both x and y are on the same side of the equation. These two equations are of the form  We substituted  to find the x- intercept and  to find the y-intercept, and then found a third point by choosing another value for x or y.
Equation #5 is written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.
This leads to the following strategy.

### Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real world situations.
Usually, when a linear equation models uses real-world data, different letters are used for the variables, instead of using only x and y. The variable names remind us of what quantities are being measured.
Also, we often will need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.
The cost of running some types of business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

### Use Slopes to Identify Parallel and Perpendicular Lines

Two lines that have the same slope are called parallel lines. Parallel lines have the same steepness and never intersect.
We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See .
Verify that both lines have the same slope,  and different y-intercepts.
What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel, like the lines shown in this graph.
Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.
Let’s look at the lines whose equations are  and  shown in .
These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.
If we look at the slope of the first line,  and the slope of the second line,  we can see that they are negative reciprocals of each other. If we multiply them, their product is
This is always true for perpendicular lines and leads us to this definition.
We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.
We find the slope–intercept form of the equation, and then see if the slopes are opposite reciprocals. If the product of the slopes is  the lines are perpendicular.

### Key Concepts

1. Slope of a Line

2. How to find the slope of a line from its graph using

3. Slope of a line between two points.

4. How to graph a line given a point and the slope.

5. Slope Intercept Form of an Equation of a Line

6. Parallel Lines

7. Perpendicular Lines

### Practice Makes Perfect

Find the Slope of a Line
In the following exercises, find the slope of each line shown.
In the following exercises, find the slope of each line.
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
In the following exercises, graph the line of each equation using its slope and y-intercept.
Choose the Most Convenient Method to Graph a Line
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Graphs and Functions

## Find the Equation of a Line

How do online companies know that “you may also like” a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.
The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section, we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.

### Find an Equation of the Line Given the Slope and y-Intercept

We can easily determine the slope and intercept of a line if the equation is written in slope-intercept form,  Now we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.
Sometimes, the slope and intercept need to be determined from the graph.

### Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope-intercept form of the equation works well when you are given the slope and y-intercept or when you read them off a graph. But what happens when you have another point instead of the y-intercept?
We are going to use the slope formula to derive another form of an equation of the line.
Suppose we have a line that has slope m and that contains some specific point  and some other point, which we will just call  We can write the slope of this line and then change it to a different form.
This format is called the point-slope form of an equation of a line.
We can use the point-slope form of an equation to find an equation of a line when we know the slope and at least one point. Then, we will rewrite the equation in slope-intercept form. Most applications of linear equations use the the slope-intercept form.
We list the steps for easy reference.

### Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.
So far, we have two options for finding an equation of a line: slope-intercept or point-slope. When we start with two points, it makes more sense to use the point-slope form.
But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.
The steps are summarized here.
We have seen that we can use either the slope-intercept form or the point-slope form to find an equation of a line. Which form we use will depend on the information we are given.

### Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point-slope equation.
First, let’s look at this graphically.
This graph shows  We want to graph a line parallel to this line and passing through the point
We know that parallel lines have the same slope. So the second line will have the same slope as  That slope is  We’ll use the notation  to represent the slope of a line parallel to a line with slope m. (Notice that the subscript || looks like two parallel lines.)
The second line will pass through  and have
To graph the line, we start at and count out the rise and run.
With  (or ), we count out the rise 2 and the run 1. We draw the line, as shown in the graph.
Do the lines appear parallel? Does the second line pass through
We were asked to graph the line, now let’s see how to do this algebraically.
We can use either the slope-intercept form or the point-slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point-slope form.

### Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point-slope equation, like we did with parallel lines.
This graph shows  Now, we want to graph a line perpendicular to this line and passing through
We know that perpendicular lines have slopes that are negative reciprocals.
We’ll use the notation  to represent the slope of a line perpendicular to a line with slope m. (Notice that the subscript  looks like the right angles made by two perpendicular lines.)
We now know the perpendicular line will pass through  with
To graph the line, we will start at  and count out the rise  and the run 2. Then we draw the line.
Do the lines appear perpendicular? Does the second line pass through
We were asked to graph the line, now, let’s see how to do this algebraically.
We can use either the slope-intercept form or the point-slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point-slope form.
In , we used the point-slope form to find the equation. We could have looked at this in a different way.
We want to find a line that is perpendicular to  that contains the point  This graph shows us the line and the point
We know every line perpendicular to a vertical line is horizontal, so we will sketch the horizontal line through
Do the lines appear perpendicular?
If we look at a few points on this horizontal line, we notice they all have y-coordinates of  So, the equation of the line perpendicular to the vertical line  is

### Key Concepts

1. How to find an equation of a line given the slope and a point.

2. How to find an equation of a line given two points.

3. How to find an equation of a line parallel to a given line.

4. How to find an equation of a line perpendicular to a given line.

### Practice Makes Perfect

Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope-intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope-intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form.
Mixed Practice
In the following exercises, find the equation of each line. Write the equation in slope-intercept form.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Graphs and Functions

## Graph Linear Inequalities in Two Variables

### Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.
Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.
Recall that an inequality with one variable had many solutions. For example, the solution to the inequality  is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See .

Similarly, linear inequalities in two variables have many solutions. Any ordered pair  that makes an inequality true when we substitute in the values is a solution to a linear inequality.

### Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.
Let’s think about the number line in shown previously again. The point  separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See .
Similarly, the line  separates the plane into two regions. On one side of the line are points with  On the other side of the line are the points with  We call the line  a boundary line.
For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:
Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.
Now, let’s take a look at what we found in . We’ll start by graphing the line  and then we’ll plot the five points we tested, as shown in the graph. See .
In  we found that some of the points were solutions to the inequality  and some were not.
Which of the points we plotted are solutions to the inequality
The points  and  are solutions to the inequality  Notice that they are both on the same side of the boundary line
The two points  and  are on the other side of the boundary line  and they are not solutions to the inequality  For those two points,
What about the point  Because  the point is a solution to the equation  but not a solution to the inequality  So the point  is on the boundary line.
Let’s take another point above the boundary line and test whether or not it is a solution to the inequality  The point  clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?
So,  is a solution to
Any point you choose above the boundary line is a solution to the inequality  All points above the boundary line are solutions.
Similarly, all points below the boundary line, the side with  and  are not solutions to  as shown in .
The graph of the inequality  is shown in below.
The line  divides the plane into two regions. The shaded side shows the solutions to the inequality
The points on the boundary line, those where  are not solutions to the inequality  so the line itself is not part of the solution. We show that by making the line dashed, not solid.

### Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.
The steps we take to graph a linear inequality are summarized here.
What if the boundary line goes through the origin? Then, we won’t be able to use  as a test point. No problem—we’ll just choose some other point that is not on the boundary line.
Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line.
Recall that:

### Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

### Key Concepts

1. How to graph a linear inequality in two variables.

### Practice Makes Perfect

Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities in Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Graphs and Functions

## Relations and Functions

### Find the Domain and Range of a Relation

As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.
When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation.
There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.
A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all x-coordinates is the domain of the relation and the set of all y-coordinates is the range. Generally we write the numbers in ascending order for both the domain and range.

### Determine if a Relation is a Function

A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
The birthday example from  helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24th as their birthday and that June and Liz share August 2nd. Since each person has exactly one birthday, the relation in  is a function.
The relation shown by the graph in  includes the ordered pairs  and  Is that okay in a function? No, as this is like one person having two different birthdays.
In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for y. If each value of x results in only one value of y, then the equation defines a function.

### Find the Value of a Function

It is very convenient to name a function and most often we name it f, g, h, F, G, or H. In any function, for each x-value from the domain we get a corresponding y-value in the range. For the function f, we write this range value y as  This is called function notation and is read f of x or the value of f at x. In this case the parentheses does not indicate multiplication.
We call x the independent variable as it can be any value in the domain. We call y the dependent variable as its value depends on x.
Much as when you first encountered the variable x, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.
Let’s look at the equation  To find the value of y when  we know to substitute  into the equation and then simplify.
The value of the function at  is 3.
We do the same thing using function notation, the equation  can be written as  To find the value when  we write:
The value of the function at  is 3.
This process of finding the value of  for a given value of x is called evaluating the function.
In the last example, we found  for a constant value of x. In the next example, we are asked to find  with values of x that are variables. We still follow the same procedure and substitute the variables in for the x.
Many everyday situations can be modeled using functions.

### Key Concepts

1. Function Notation: For the function

2. Independent and Dependent Variables: For the function

### Practice Makes Perfect

Find the Domain and Range of a Relation
In the following exercises, for each relation ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercises, use the mapping of the relation to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function, ⓑ find the domain of the function, and ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function: ⓐ  ⓑ  ⓒ
In the following exercises, evaluate the function: ⓐ  ⓑ  ⓒ
In the following exercises, evaluate the function.
In the following exercises, solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?