# Foundations

## Decimals

### Name and Write Decimals

Decimals are another way of writing fractions whose denominators are powers of 10.
Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.
When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values.  shows the names of the place values to the left and right of the decimal point.
We summarize the steps needed to name a decimal below.
When we write a check we write both the numerals and the name of the number. Let’s see how to write the decimal from the name.
We summarize the steps to writing a decimal.

### Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.
We summarize the steps for rounding a decimal here.

### Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

### Multiply and Divide Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first convert them to fractions and then multiply.
So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern!
Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.
We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.
The rules for multiplying positive and negative numbers apply to decimals, too, of course!
When multiplying two numbers,
1. if their signs are the same the product is positive.
2. if their signs are different the product is negative.
When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.
In many of your other classes, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.
Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
To divide decimals, determine what power of 10 to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of  Because of the equivalent fractions property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example:
We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
We review the notation and vocabulary for division:
We’ll write the steps to take when dividing decimals, for easy reference.
A common application of dividing whole numbers into decimals is when we want to find the price of one item that is sold as part of a multi-pack. For example, suppose a case of 24 water bottles costs $3.99. To find the price of one water bottle, we would divide $3.99 by 24. We show this division in . In calculations with money, we will round the answer to the nearest cent (hundredth).

### Convert Decimals, Fractions, and Percents

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.
Notice, when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator. Fractions like this are called proper fractions.
The steps to take to convert a decimal to a fraction are summarized in the procedure box.
We’ve learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar means division. So  can be written  or  This leads to the following method for converting a fraction to a decimal.
When we divide, we will not always get a zero remainder. Sometimes the quotient ends up with a decimal that repeats. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating block of digits to indicate it repeats.
A bar is placed over the repeating block of digits to indicate it repeats.
Sometimes we may have to simplify expressions with fractions and decimals together.
A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent.
Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator.
Do you see the pattern? To convert a percent number to a decimal number, we move the decimal point two places to the left.
Converting a decimal to a percent makes sense if we remember the definition of percent and keep place value in mind.
To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.
Recognize the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

### Key Concepts

1. Name a Decimal

2. Write a Decimal

3. Round a Decimal

4. Add or Subtract Decimals

5. Multiply Decimals

6. Multiply a Decimal by a Power of Ten

7. Divide Decimals

8. Convert a Decimal to a Proper Fraction

9. Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

### Practice Makes Perfect

Name and Write Decimals
In the following exercises, write as a decimal.
In the following exercises, name each decimal.
Round Decimals
In the following exercises, round each number to the nearest tenth.
In the following exercises, round each number to the nearest hundredth.
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide.
Convert Decimals, Fractions and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.

### 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?


# Foundations

## The Real Numbers

### Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write  and read it “n squared.” The result is called the square of n. For example,
Similarly, 121 is the square of 11, because  is 121.
Complete the following table to show the squares of the counting numbers 1 through 15.
The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.
The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.
Did you notice that these squares are the same as the squares of the positive numbers?
Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because  we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is  is called a square root of m.
Notice  also, so  is also a square root of 100. Therefore, both 10 and  are square roots of 100.
So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign,  denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.
We also use the radical sign for the square root of zero. Because   Notice that zero has only one square root.
Since 10 is the principal square root of 100, we write  You may want to complete the following table to help you recognize square roots.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,  We read  as “the opposite of the square root of 10.”

### Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as , , and . What is the difference between these types of numbers?
What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
All signed fractions, such as  are rational numbers. Each numerator and each denominator is an integer.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to
An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.
Since any integer can be written as the ratio of two integers, all integers are rational numbers! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.
We’ve already seen that integers are rational numbers. The integer  could be written as the decimal  So, clearly, some decimals are rational.
Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means  we can write it as an improper fraction,  So 7.3 is the ratio of the integers 73 and 10. It is a rational number.
In general, any decimal that ends after a number of digits (such as 7.3 or  is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.
Let’s look at the decimal form of the numbers we know are rational.
We have seen that every integer is a rational number, since  for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.
We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we considered above.
What do these examples tell us?
Every rational number can be written both as a ratio of integers,  where p and q are integers and  and as a decimal that either stops or repeats.
Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:
Are there any decimals that do not stop or repeat? Yes!
The number  (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.
We can even create a decimal pattern that does not stop or repeat, such as
Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.
Let’s summarize a method we can use to determine whether a number is rational or irrational.
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
All the numbers we use in elementary algebra are real numbers.  illustrates how the number sets we’ve discussed in this section fit together.
Can we simplify  Is there a number whose square is
None of the numbers that we have dealt with so far has a square that is  Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to
The square root of a negative number is not a real number.

### Locate Fractions on the Number Line

The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.
Let’s start with fractions and locate  on the number line.
We’ll start with the whole numbers  and  because they are the easiest to plot. See .
The proper fractions listed are  We know the proper fraction  has value less than one and so would be located between  The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts  We plot  See .
Similarly,  is between 0 and  After dividing the unit into 5 equal parts we plot  See .
Finally, look at the improper fractions  These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See .
 shows the number line with all the points plotted.
In , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.
1. a < b “a is less than b” when a is to the left of b on the number line
2. a > b “a is greater than b” when a is to the right of b on the number line
As we move from left to right on a number line, the values increase.

### Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.
Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

Again, we can use the number line to order numbers.
1. a < b “a is less than b” when a is to the left of b on the number line
2. a > b “a is greater than b” when a is to the right of b on the number line
Where are 0.04 and 0.40 located on the number line? See .
We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.
How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.
Because 310 > 308, we know that  Therefore, 0.31 > 0.308.
Notice what we did in converting 0.31 to a fraction—we started with the fraction  and ended with the equivalent fraction  Converting  back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!
We say 0.31 and 0.310 are equivalent decimals.
We use equivalent decimals when we order decimals.
The steps we take to order decimals are summarized here.
When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because  lies to the right of  on the number line, we know that  Similarly, smaller numbers lie to the left on the number line. For example, because  lies to the left of  on the number line, we know that  See .
If we zoomed in on the interval between 0 and  as shown in , we would see in the same way that

### Key Concepts

1. Square Root Notation
 is read ‘the square root of m.’ If  then  for
2. Order Decimals

### Practice Makes Perfect

Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers
In the following exercises, identify whether each number is rational or irrational.
In the following exercises, identify whether each number is a real number or not a real number.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
In the following exercises, order each of the pairs of numbers, using < or >.
Locate Decimals on the Number Line In the following exercises, locate the number on the number line.
In the following exercises, order each pair of numbers, using < or >.

### Everyday Math

### Writing Exercises

### Self Check

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


# Foundations

## Properties of Real Numbers

### Use the Commutative and Associative Properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?
The results are the same.
As we can see, the order in which we add does not matter!
What about multiplying
Again, the results are the same!
The order in which we multiply does not matter!
These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.
The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
What about subtraction? Does order matter when we subtract numbers? Does  give the same result as
The results are not the same.
Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
The results are not the same.
Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!
1. Addition and multiplication are commutative.
2. Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?
Some people would think  and then  Others might start with  and then
Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.
When adding three numbers, changing the grouping of the numbers gives the same result.
This is true for multiplication, too.
When multiplying three numbers, changing the grouping of the numbers gives the same result.
You probably know this, but the terminology may be new to you. These examples illustrate the associative property.
Let’s think again about multiplying  We got the same result both ways, but which way was easier? Multiplying  and  first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!
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 or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.

### Use the Identity and Inverse Properties of Addition and Multiplication

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.
For example,
These examples illustrate 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.
For example,
These examples illustrate the Identity Property of Multiplication that states that for any real number   and
We summarize the Identity Properties below.

Notice that in each case, the missing number was the opposite of the number!
We call  the additive inverse of a. 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  Remember, a number and its opposite add to zero.
What number multiplied by  gives the multiplicative identity, 1? In other words,  times what results in 1?
What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?
Notice that in each case, the missing number was the reciprocal of the number!
We call  the multiplicative inverse of a. The reciprocal of  A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number
We’ll formally state the inverse properties here:

### Use the Properties of Zero

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 0? Think about the related multiplication fact:  means  Is there a number that multiplied by 0 gives 4? 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 below.
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.

### 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.
Back to our friends at the movies, we could find the total amount of money they need like this:
In algebra, we use the distributive property to remove parentheses as we simplify expressions.
For example, if we are asked to simplify the expression  the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in .
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  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!
 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 in .

### Key Concepts

1. Commutative Property of

2. Associative Property of

3. Distributive Property: If  are real numbers, then

4. Identity Property

5. Inverse Property

6. Properties of Zero

### Practice Makes Perfect

Use the Commutative and Associative Properties
In the following exercises, use the associative property to simplify.
In the following exercises, simplify.
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
In the following exercises, find the multiplicative inverse of each number.
Use the Properties of Zero
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the distributive property.

### Everyday Math

### 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?


# Foundations

## Systems of Measurement

### Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.
The equivalencies of measurements are shown in . The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.
To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction  When we multiply by this fraction we do not change the value, but just change the units.
But  also equals 1. How do we decide whether to multiply by  or  We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert  inches to feet, which multiplication will eliminate the inches?
The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.
When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

### Use Mixed Units of Measurement in the U.S. System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.
Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

### Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is  of a meter, just like one cent is  of one dollar.
The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.
To make conversions in the metric system, we will use the same technique we did in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a 5K or 10K race? The length of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.
As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.
To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.
We can apply this pattern when we make measurement conversions in the metric system. In , we changed 3,200 grams to kilograms by multiplying by  (or 0.001). This is the same as moving the decimal three places to the left.

### Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.

### Convert Between the U.S. and the Metric Systems of Measurement

Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.
 shows some of the most common conversions.
 shows how inches and centimeters are related on a ruler.
 shows the ounce and milliliter markings on a measuring cup.
 shows how pounds and kilograms marked on a bathroom scale.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

### Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for  what does that mean?
The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written  The metric system uses degrees Celsius, written   shows the relationship between the two systems.

### Key Concepts

1. Metric System of Measurement

2. Temperature Conversion

### Section Exercises

### Practice Makes Perfect

Make Unit Conversions in the U.S. System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the U.S. System
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exercises, solve.
Convert Between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Everyday Math

### Writing Exercises

### Self Check

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

### Chapter Review Exercises

### Introduction to Whole Numbers

Use Place Value with Whole Number
In the following exercises find the place value of each digit.
In the following exercises, name each number.
In the following exercises, write each number as a whole number using digits.
In the following exercises, round to the indicated place value.
In the following exercises, round each number to the nearest ⓐ hundred ⓑ thousand ⓒ ten thousand.
Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of the following numbers using the multiples method.
In the following exercises, find the least common multiple of the following numbers using the prime factors method.

### Use the Language of Algebra

Use Variables and Algebraic Symbols
In the following exercises, translate the following from algebra to English.
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
In the following exercises, simplify
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
In the following exercises, identify the terms in each expression.
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 following phrases into algebraic expressions.

### Add and Subtract Integers

Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
In the following exercises,, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions with Absolute Value
In the following exercises,, simplify.
In the following exercises, fill in <, >, or = for each of the following pairs of numbers.
In the following exercises, simplify.
In the following exercises, evaluate.
Add Integers
In the following exercises, simplify each expression.
Subtract Integers
In the following exercises, simplify.
In the following exercises, simplify each expression.
Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In 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 exercises, solve.

### Visualize Fractions

Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
Simplify Fractions
In the following exercises, simplify.
Multiply Fractions
In the following exercises, multiply.
Divide Fractions
In the following exercises, divide.
In the following exercises, simplify.
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.

### Add and Subtract Fractions

Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.

### Decimals

Name and Write Decimals
In the following exercises, write as a decimal.
In the following exercises, name each decimal.
Round Decimals
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide.
Convert Decimals, Fractions, and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.

### The Real Numbers

Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers.
In the following exercises, identify whether each number is rational or irrational.
In the following exercises, identify whether each number is a real number or not a real number.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
In the following exercises, order each of the following pairs of numbers, using < or >.
Locate Decimals on the Number Line
In the following exercises, locate on the number line.
In the following exercises, order each of the following pairs of numbers, using < or >.

### Properties of Real Numbers

Use the Commutative and Associative Properties
In the following exercises, use the Associative Property to simplify.
In the following exercises, simplify.
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
In the following exercises, find the multiplicative inverse of each number.
Use the Properties of Zero
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Systems of Measurement

1.1 Define U.S. Units of Measurement and Convert from One Unit to Another
In the following exercises, convert the units. Round to the nearest tenth.
Use Mixed Units of Measurement in the U.S. System.
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exerices, solve.
Convert between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Chapter Practice Test

In the following exercises, evaluate.
In the following exercises, simplify each expression.


# Solving Linear Equations and Inequalities

## Introduction

If we carefully placed more rocks of equal weight on both sides of this formation, it would still balance.  Similarly, the expressions in an equation remain balanced when we add the same quantity to both sides of the equation.  In this chapter, we will solve equations, remembering that what we do to one side of the equation, we must also do to the other side.


# Solving Linear Equations and Inequalities

## Solve Equations Using the Subtraction and Addition Properties of Equality

### Verify a Solution of an Equation

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 make each side of the equation the same – so that we end up with 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!

### Solve Equations Using the Subtraction and Addition Properties of Equality

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in . Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?
What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?
Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See  for an illustration of this process.
What algebraic equation would match this situation? In  each side of the workspace represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope .
Let’s write algebraically the steps we took to discover how many counters were in the envelope:
Check:
Five in the envelope plus three more does equal eight!
Our model has given us an idea of what we need to do to solve one kind of equation. The goal is to isolate the variable by itself on one side of the equation. To solve equations such as these mathematically, we use the Subtraction Property of Equality.
Let’s see how to use this property to solve an equation. Remember, the goal is to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.
What happens when an equation has a number subtracted from the variable, as in the equation ? We use another property of equations to solve equations where a number is subtracted from the variable. We want to isolate the variable, so to ‘undo’ the subtraction we will add the number to both sides. We use the Addition Property of Equality.
In , 37 was added to the y and so we subtracted 37 to ‘undo’ the addition. In , we will need to ‘undo’ subtraction by using the Addition Property of Equality.
The next example will be an equation with decimals.

### Solve Equations That Require Simplification

In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.
You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.

### Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign.  shows us some of the words that are commonly used.
The steps we use to translate a sentence into an equation are listed below.

### Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.
We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.

### Key Concepts

1. To Determine Whether a Number is a Solution to an Equation

2. Addition Property of Equality

3. Subtraction Property of Equality

4. To Translate a Sentence to an Equation

5. To Solve an Application

### Practice Makes Perfect

Verify a Solution of an Equation
In the following exercises, determine whether the given value is a solution to the equation.
Solve Equations using the Subtraction and Addition Properties of Equality
In the following exercises, solve each equation using the Subtraction and Addition Properties of Equality.
Solve Equations that Require Simplification
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve it.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.

### 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 your goals 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 as topics you do not master become potholes in your road to success. Math is sequential - 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 critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.


# Solving Linear Equations and Inequalities

## Solve Equations using the Division and Multiplication Properties of Equality

### Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form  or . We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.
Let’s look at our puzzle again with the envelopes and counters in .
In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since ).
What equation models the situation shown in ? There are two envelopes, and each contains  counters. Together, the two envelopes must contain a total of 6 counters.
We found that each envelope contains 3 counters. Does this check? We know , so it works! Three counters in each of two envelopes does equal six!
This example leads to the Division Property of Equality.
The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.
Consider the equation . We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

### Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.
With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.
Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

### Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

### Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

### Key Concepts

1. The Division Property of Equality—For any numbers a, b, and c, and , if , then .
When you divide both sides of an equation by any non-zero number, you still have equality.
2. The Multiplication Property of Equality—For any numbers a, b, and c, if , then .
If you multiply both sides of an equation by the same number, you still have equality.

### Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Mixed Practice
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and 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 and Inequalities

## Solve Equations with Variables and Constants on Both Sides

### Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.
Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.
By doing this, we will transform the equation that began with variables and constants on both sides into the form  We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

### Solve Equations with Variables on Both Sides

What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

### Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.
We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.
In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.
In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!
To solve an equation with fractions, we just follow the steps of our strategy to get the solution!
We will use the same strategy to find the solution for an equation with decimals.

### Key Concepts

1. Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation

### Practice Makes Perfect

Solve Equations with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve Equations with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve Equations with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.

### 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 and Inequalities

## Use a General Strategy to Solve Linear Equations

### Solve Equations Using the General Strategy

Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.
Beginning by simplifying each side of the equation makes the remaining steps easier.

### Classify Equations

Consider the equation we solved at the start of the last section, . The solution we found was . This means the equation  is true when we replace the variable, x, with the value . We showed this when we checked the solution  and evaluated  for .
If we evaluate  for a different value of x, the left side will not be .
The equation  is true when we replace the variable, x, with the value , but not true when we replace x with any other value. Whether or not the equation  is true depends on the value of the variable. Equations like this are called conditional equations.
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.
But  is true.
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 like this is called an identity.
What happens when we solve the equation ?
But .
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.

### Key Concepts

1. General Strategy for Solving Linear Equations

### Practice Makes Perfect

Solve Equations Using the General Strategy for Solving Linear Equations
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.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective 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 and Inequalities

## Solve Equations with Fractions or Decimals

### Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, .
This method worked 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 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.
Let’s solve a similar equation, but this time use the method that eliminates the fractions.
Notice in , 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 again have variables on both sides of the equation.
In the next example, we start by using the Distributive Property. This step clears the fractions right away.
In the next example, even after distributing, we still have fractions to clear.

### Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will 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 money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

### Key Concepts

1. Strategy to Solve an Equation with Fraction Coefficients

### Practice Makes Perfect

Solve Equations with Fraction Coefficients
In the following exercises, solve each equation with fraction coefficients.
Solve Equations with Decimal 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.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Solving Linear Equations and Inequalities

## Solve a Formula for a Specific Variable

### Use the Distance, Rate, and Time Formula

One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive at a steady rate of 60 miles per hour for 2 hours? (This might happen if you use your car’s cruise control while driving on the highway.) If you said 120 miles, you already know how to use this formula!
We will use the Strategy for Solving Applications that we used earlier in this chapter. When our problem requires a formula, we change Step 4. In place of writing a sentence, we write the appropriate formula. We write the revised steps here for reference.
You may want to create a mini-chart to summarize the information in the problem. See the chart in this first example.

### Solve a Formula for a Specific Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.
In  and , we used the formula . This formula gives the value of , distance, when you substitute in the values of , the rate and time. But in , we had to find the value of . We substituted in values of  and then used algebra to solve for . If you had to do this often, you might wonder why there is not a formula that gives the value of  when you substitute in the values of . We can make a formula like this by solving the formula  for .
To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.
The formula  is used to calculate simple interest, I, for a principal, P, invested at rate, r, for t years.
Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually x and y. You 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.
In Examples 1.60 through 1.64 we used the numbers in part ⓐ as a guide to solving in general in part ⓑ. Now we will solve a formula in general without using numbers as a guide.

### Key Concepts

1. To Solve an Application (with a formula)

2. Distance, Rate and Time
For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:  where d = distance, r = rate, t = time.
3. To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

### Practice Makes Perfect

Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, use the formula .
In the following exercises, use the formula .
In the following exercises, use the formula I = Prt.
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 and Inequalities

## Solve Linear Inequalities

### Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
What about the solution of an inequality? What number would make the inequality  true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality .
We show the solutions to the inequality  on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of  is shown in . Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.
The graph of the inequality  is very much like the graph of , but now we need to show that 3 is a solution, too. We do that by putting a bracket at , as shown in .
Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.
We can also represent inequalities using interval notation. As we saw above, the inequality  means all numbers greater than 3. 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.
The inequality  means all numbers less than or equal to 1. 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.
Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in .

### Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.
Similar properties hold true for inequalities.
Similarly we could show that the inequality also stays the same for addition.
This leads us to the Subtraction and Addition Properties of Inequality.
We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality , the steps would look like this:
Any number greater than 4 is a solution to this inequality.

### Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Consider some numerical examples.
Does the inequality stay the same when we divide or multiply by a negative number?
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.
Here are the Division and Multiplication Properties of Inequality for easy reference.
When we divide or multiply an inequality by a:
1. positive number, the inequality stays the same.
2. negative number, the inequality reverses.

### Solve Inequalities That Require Simplification

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 be sure to pay close attention during multiplication or division.
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.

### 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.
Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.
 shows some common phrases that indicate inequalities.

### Key Concepts

1. Subtraction Property of Inequality
For any numbers a, b, and c,
if  then  and
if  then
2. Addition Property of Inequality
For any numbers a, b, and c,
if  then  and
if  then
3. Division and Multiplication Properties of Inequality
For any numbers a, b, and c,
if  and , then  and .
if  and , then  and .
if  and , then  and .
if  and , then  and .
4. When we divide or multiply an inequality by a:

### Section Exercises

### Practice Makes Perfect

Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
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.
Translate 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.

### 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?

### Chapter 2 Review Exercises

### Solve Equations using the Subtraction and Addition Properties of Equality

Verify a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
Solve Equations using the Subtraction and Addition Properties of Equality
In the following exercises, solve each equation using the Subtraction Property of Equality.
In the following exercises, solve each equation using the Addition Property of Equality.
In the following exercises, solve each equation.
Solve Equations That Require Simplification
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Translate and Solve Applications
In the following exercises, translate into an algebraic equation and solve.

### Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.

### Solve Equations with Variables and Constants on Both Sides

Solve an Equation with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve an Equation with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.

### Use a General Strategy for Solving Linear Equations

Solve Equations Using the General Strategy for Solving Linear Equations
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 Fractions and Decimals

Solve Equations with Fraction Coefficients
In the following exercises, solve each equation with fraction coefficients.
Solve Equations with Decimal Coefficients
In the following exercises, solve each equation with decimal coefficients.

### Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, solve.

### Solve Linear Inequalities

Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
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 write the solution in interval notation and graph on the number line.

### Everyday Math

### Chapter 2 Practice Test

In the following exercises, solve each equation.
In the following exercises, graph 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.


# Math Models

## Introduction

Mathematical formulas model phenomena in every facet of our lives. They are used to explain events and predict outcomes in fields such as transportation, business, economics, medicine, chemistry, engineering, and many more. In this chapter, we will apply our skills in solving equations to solve problems in a variety of situations.


# Math Models

## Use a Problem-Solving Strategy

### Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in  and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

### Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
Let’s try this approach with another example.

### Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
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. So 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
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 2 more than the number preceding it. If we call the first one n, then the next one is  The next one would be  or
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.
Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you 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 don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

### Key Concepts

1. Problem-Solving Strategy

2. Consecutive Integers
Consecutive integers are integers that immediately follow each other.

Consecutive even integers are even integers that immediately follow one another.

Consecutive odd integers are odd integers that immediately follow one another.

### Practice Makes Perfect

Use the Approach Word Problems with a Positive Attitude
In the following exercises, prepare the lists described.
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 Problems
In the following exercises, solve each number word problem.

### 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 your goals 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 as topics you do not master become potholes in your road to success. Math is sequential—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 critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.


# Math Models

## Solve Percent Applications

### Translate and Solve Basic Percent Equations

We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.
We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.
We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in . We will again use direct translation to write the equation.
In the next example, we are looking for the percent.

### Solve Applications of Percent

Many applications of percent—such as tips, sales tax, discounts, and interest—occur in our daily lives. To solve these applications we’ll translate to a basic percent equation, just like those we solved in previous examples. Once we translate the sentence into a percent equation, we know how to solve it.
We will restate the problem solving strategy we used earlier for easy reference.
Now that we have the 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 will involve everyday situations, you can rely on your own experience.

### Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.
To find the percent increase, first we find the amount of increase, the difference of the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.
Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference of the original amount and the new amount. Then we find what percent the amount of decrease is of the original amount.

### Solve Simple Interest Applications

Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, P, and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; called the rate of interest, r. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is in the account.
To find the interest we use the simple interest formula,
Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.
The formula we use to calculate simple interest is  To use the formula, we substitute in the values the problem gives us for the variables, 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 don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.
Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.

### Solve Applications with Discount or Mark-up

Applications of discount 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.
We summarize the discount model in the box below.
There may be times when we know the original price and the sale price, and we want to know the discount rate. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price.  will show this case.
Applications of mark-up are very common in retail settings. 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.
We summarize the mark-up model in the box below.

### Key Concepts

1. Percent Increase To find the percent increase:

2. Percent Decrease To find the percent decrease:

3. Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is

4. Discount

5. Mark-up

### Practice Makes Perfect

Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Percent Applications
In the following exercises, solve.
Find Percent Increase and Percent Decrease
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
Solve Applications with Discount or Mark-up
In the following exercises, find the sale price.
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.

### Everyday Math

### 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 goals?


# Math Models

## Solve Mixture Applications

### Solve Coin Word Problems

In mixture problems, we will have two or more items with different values to combine together. The mixture model is used by grocers and bartenders to make sure they set fair prices for the products they sell. Many other professionals, like chemists, investment bankers, and landscapers also use the mixture model.
We will start by looking at an application everyone is familiar with—money!
Imagine that we take a handful of coins from a pocket or purse and place them on a desk. How would we determine the value of that pile of coins? If we can form a step-by-step plan for finding the total value of the coins, it will help us as we begin solving coin word problems.
So what would we do? To get some order to the mess of coins, we could separate the coins into piles according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, we would add the total value of each pile.
How would we determine the value of each pile? Think about the dime pile—how much is it worth? 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 17 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 17 dimes, multiply 17 by $0.10 to get $1.70. This is the total value of all 17 dimes. This method leads to the following model.
The number of dimes times the value of each dime equals the total value of the dimes.
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.
Let’s look at a specific case. Suppose there are 14 quarters, 17 dimes, 21 nickels, and 39 pennies.
The total value of all the coins is $6.64.
Notice how the chart helps organize all the information! Let’s see how we use this method to solve a coin word problem.
In the next example, we’ll show only the completed table—remember the steps we take to fill in the table.

### 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.
We have learned how to find the total number of tickets when the number of one type of ticket is based on the number of the other type. Next, we’ll look at an example where we know the total number of tickets and have to figure out how the two types of tickets relate.
Suppose Bianca sold a total of 100 tickets. Each ticket was either an adult ticket or a child ticket. If she sold 20 child tickets, how many adult tickets did she sell?
1. Did you say ‘80’? How did you figure that out? Did you subtract 20 from 100?
If she sold 45 child tickets, how many adult tickets did she sell?
1. Did you say ‘55’? How did you find it? By subtracting 45 from 100?
What if she sold 75 child tickets? How many adult tickets did she sell?
1. The number of adult tickets must be
Now, suppose Bianca sold x child tickets. Then how many adult tickets did she 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 below.
We can apply these techniques to other examples
Now, we’ll do one where we fill in the table all at once.

### Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. Grocers and bartenders use the mixture model to set a fair price for a product made from mixing two or more ingredients. Financial planners use the mixture model when they invest money in a variety of accounts and want to find the overall interest rate. Landscape designers use the mixture model when they have an assortment of plants and a fixed budget, and event coordinators do the same when choosing appetizers and entrees for a banquet.
Our first mixture word problem will be making trail mix from raisins and nuts.
We can also use the mixture model to solve investment problems using simple interest. We have used the simple interest formula,  where  represented the number of years. When we just need to find the interest for one year,  so then

### Key Concepts

1. Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
 where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
2. Problem-Solving Strategy—Coin Word Problems

### 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.

### Everyday Math

### 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?