# Prerequisites

## Real Numbers: Algebra Essentials

### Learning Objectives

1. Identify the study skills leading to success in a college level mathematics course.
2. Reflect on your past math experiences and create a plan for improvement.

### Objective 1: Identify the study skills leading to success in a college level mathematics course.

Welcome to your algebra course! This course will be challenging so now is the time to set up a plan for success. In this first chapter we will focus on important strategies for success including: math study skills, time management, note taking skills, smart test taking strategies, and the idea of a growth mindset. Each of these ideas will help you to be successful in your college level math course whether you are enrolled in a face-to-face traditional section or an online section virtual section.

Complete the following survey by checking a column for each behavior based on the frequency that you engage in the behavior during your last academic term.

### Practice Makes Perfect

Identify the study skills leading to success in a college level mathematics course.

### Objective 2: Reflect on your past math experiences and create a plan for improvement.

1.
It’s important to take the opportunity to reflect on your past experiences in math classes as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you had in past math courses and list a possible solution that you could try this semester.

2.
Write your math autobiography. Tell your math story by describing your past experiences as a learner of mathematics. Share how your attitudes have changed about math over the years if they have. Perhaps include what you love, hate, dread, appreciate, fear, look forward to, or find beauty in. This will help your teacher to better understand you and your current feelings about the discipline.

3.
Share your autobiographies with your study group members. This helps to create a community in the classroom when common themes emerge.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.
Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.
But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.
Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.
Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

### Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as
    where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero:

The set of integers adds the opposites of the natural numbers to the set of whole numbers:
    It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.
The set of rational numbers is written as
    Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.
Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:
1. ⓐa terminating decimal:
    or
2. ⓑa repeating decimal:

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

### Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even
    but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

### Real Numbers

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in .

### Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as .

### Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example,
    We can raise any number to any power. In general, the exponential notation
    means that the number or variable
    is used as a factor
    times.
In this notation,
    is read as the nth power of
   or  to the  where
    is called the base and
    is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example,
    is a mathematical expression.
To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.
The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.
Let’s take a look at the expression provided.
There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify
    as 16.
Next, perform multiplication or division, left to right.
Lastly, perform addition or subtraction, left to right.
Therefore,

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

### Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

### Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.
We can better see this relationship when using real numbers.
Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
Again, consider an example with real numbers.
It is important to note that neither subtraction nor division is commutative. For example,
    is not the same as
    Similarly,

### Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.
Consider this example.
The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
This property can be especially helpful when dealing with negative integers. Consider this example.
Are subtraction and division associative? Review these examples.
As we can see, neither subtraction nor division is associative.

### Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.
This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.
Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.
To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.
A special case of the distributive property occurs when a sum of terms is subtracted.
For example, consider the difference
    We can rewrite the difference of the two terms 12 and
    by turning the subtraction expression into addition of the opposite. So instead of subtracting
    we add the opposite.
Now, distribute
    and simplify the result.
This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

### Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.
The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.
For example, we have
    and
    There are no exceptions for these properties; they work for every real number, including 0 and 1.

### Inverse Properties

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by (−a), that, when added to the original number, results in the additive identity, 0.
For example, if
    the additive inverse is 8, since

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted
    that, when multiplied by the original number, results in the multiplicative identity, 1.
For example, if
    the reciprocal, denoted
    is

  because

### Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as
    or
    In the expression
    5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.
Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

### Formulas

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation
    has the solution of 3
   because when we substitute 3 for
    in the equation, we obtain the true statement

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area
    of a circle in terms of the radius
    of the circle:
    For any value of
    the area
    can be found by evaluating the expression

### Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

### Key Concepts

1. Rational numbers may be written as fractions or terminating or repeating decimals. See  and .
2. Determine whether a number is rational or irrational by writing it as a decimal. See .
3. The rational numbers and irrational numbers make up the set of real numbers. See . A number can be classified as natural, whole, integer, rational, or irrational. See .
4. The order of operations is used to evaluate expressions. See .
5. The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See .
6. Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See . They take on a numerical value when evaluated by replacing variables with constants. See , , and
7. Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See  and .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify the given expression.

### Algebraic

For the following exercises, evaluate the expression using the given value of the variable.
For the following exercises, simplify the expression.

### Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.
For the following exercises, solve the given problem.
For the following exercises, consider this scenario: There is a mound of
    pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.
For the following exercise, solve the given problem.

### Technology

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

### Extensions


# Prerequisites

## Exponents and Scientific Notation

### Learning Objective:

1. Plan your weekly academic schedule for the term.

### Objective 1: Plan your weekly academic schedule for the term.

1.
Most college instructors advocate studying at least 2 hours for each hour in class. With this recommendation in mind, complete the following table showing credit hours enrolled in, the study time required, and total time to be devoted to college work. Assume 2 hours of study time for each hour in class to complete this table, and after your first exam you can fine tune this estimate based on your performance.

Consider spending at least 2 hours of your study time each week at your campus (or virtual) math tutoring center or with a study group, the time will be well spent!

2.
Another way to optimize your class and study time is to have a plan for efficiency, meaning make every minute count. Below is a list of good practices, check off those you feel you could utilize this term.

3.
Creating your Semester Calendar- complete the following weekly schedule being sure to labelOptional: also include if you want a more comprehensive view of your time commitments

Term: ________________________________
Name: ________________________________________
Date: ________________________________________
Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per pixel, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.
Using a calculator, we enter
  and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately
  bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

### Using the Product Rule of Exponents

Consider the product
       Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.
The result is that

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.
Now consider an example with real numbers.
We can always check that this is true by simplifying each exponential expression. We find that
       is 8,
       is 16, and
       is 128. The product
       equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

### Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as
       where
       Consider the example
       Perform the division by canceling common factors.
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
For the time being, we must be aware of the condition
       Otherwise, the difference
       could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

### Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression
       The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.
The exponent of the answer is the product of the exponents:
       In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

### Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that
       so that the difference
       would never be zero or negative. What would happen if
       In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.
If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is
       This is true for any nonzero real number, or any variable representing a real number.
The sole exception is the expression
       This appears later in more advanced courses, but for now, we will consider the value to be undefined.

### Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that
       in the quotient rule even further. For example, can we simplify
       When
       —that is, where the difference
       is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.
Divide one exponential expression by another with a larger exponent. Use our example,

If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have
       This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression
       is defined when
       is a natural number, 0, or the negative of a natural number. That means that
       is defined for any integer
       Also, the product and quotient rules and all of the rules we will look at soon hold for any integer

### Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
       We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words,

### Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

### Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

### Using Scientific Notation

Recall at the beginning of the section that we found the number
  when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number,
     is positive. If you moved the decimal right as in a very small number,
     is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.
Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

### Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal
     places to the right if
     is positive or
     places to the left if
     is negative and add zeros as needed. Remember, if
     is positive, the absolute value of the number is greater than 1, and if
     is negative, the absolute value of the number is less than one.

### Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around
  molecules of water and 1 L of water holds about
  average drops. Therefore, there are approximately
  atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!
When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product
  The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as
  That adds a ten to the exponent of the answer.

### Key Equations

### Key Concepts

1. Products of exponential expressions with the same base can be simplified by adding exponents. See .
2. Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See .
3. Powers of exponential expressions with the same base can be simplified by multiplying exponents. See .
4. An expression with exponent zero is defined as 1. See .
5. An expression with a negative exponent is defined as a reciprocal. See  and .
6. The power of a product of factors is the same as the product of the powers of the same factors. See .
7. The power of a quotient of factors is the same as the quotient of the powers of the same factors. See .
8. The rules for exponential expressions can be combined to simplify more complicated expressions. See .
9. Scientific notation uses powers of 10 to simplify very large or very small numbers. See  and .
10. Scientific notation may be used to simplify calculations with very large or very small numbers. See  and .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.
For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.
For the following exercises, express the decimal in scientific notation.
For the following exercises, convert each number in scientific notation to standard notation.

### Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

### Real-World Applications

### Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

### Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.


# Prerequisites

## Radicals and Rational Exponents

### Learning Objective:

1. Investigate the discipline called learning science and the idea of a knowledge space.

### Objective 1: Investigate the discipline called learning science and the idea of a knowledge space.

The brain is a complex organ. It is the control center for our bodies, while the mind is where thinking and learning take place. In an attempt to understand the processes that occur in learning, researchers study a collection of disciplines called learning sciences. This interdisciplinary field includes study of psychological, sociological, anthropological, and computational approaches to learning.

In this skill sheet we will investigate the mathematics of mastery and knowledge spaces. A knowledge space includes the possible states of knowledge of a human learner. The theory of knowledge space was introduced in 1985 by mathematical psychologists Jean-Paul Doignon and Jean-Claude Falmagne and has since been studied by many researchers.
Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies.

### Practice Makes Perfect

Investigation:  There are 32 student-learning outcomes (SLO’s) in a typical College Algebra course. These are topics a student needs to master to show proficiency in College Algebra. Let’s begin by looking at just a few of these skills.  Let’s assign the variables, A, B, C, and D to the following topics. We will name the set containing each of these 4 topics, Q.

1. A = Graph the basic functions listed in the library of functions.
2. B = Find the domain of a function defined by an equation.
3. C = Create a new function through composition of functions.
4. D = Find linear functions that model data sets.
Using roster notation Q = {A, B, C, D}.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in , and use the Pythagorean Theorem.
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

### Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since
  the square root of

 is

 The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if

 is a positive real number, then the square root of

 is a number that, when multiplied by itself, gives

 The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals

 The square root obtained using a calculator is the principal square root.
The principal square root of

 is written as

 The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

### Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite

 as

 We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

### Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite

 as

### Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of

 and

 is

 However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression

 can be written with a

 in the radicand, as
  so

### Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is
  multiply by

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is
  then the conjugate is

### Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

### Understanding nth Roots

Suppose we know that

 We want to find what number raised to the 3rd power is equal to 8. Since
  we say that 2 is the cube root of 8.
The nth root of

 is a number that, when raised to the nth power, gives

 For example,

 is the 5th root of

 because

 If

 is a real number with at least one nth root, then the principal  of

 is the number with the same sign as

 that, when raised to the nth power, equals

The principal nth root of

 is written as
  where

 is a positive integer greater than or equal to 2. In the radical expression,

 is called the index of the radical.

### Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index

 is even, then

 cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

### Key Concepts

1. The principal square root of a number

 is the nonnegative number that when multiplied by itself equals

 See .
2. If

 and

 are nonnegative, the square root of the product

 is equal to the product of the square roots of

 and

 See  and .
3. If

 and

 are nonnegative, the square root of the quotient

 is equal to the quotient of the square roots of

 and

 See  and .
4. We can add and subtract radical expressions if they have the same radicand and the same index. See  and .
5. Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See  and .
6. The principal nth root of

 is the number with the same sign as

 that when raised to the nth power equals

 These roots have the same properties as square roots. See .
7. Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See  and .
8. The properties of exponents apply to rational exponents. See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify each expression.

### Algebraic

For the following exercises, simplify each expression.

### Real-World Applications

### Extensions

For the following exercises, simplify each expression.


# Prerequisites

## Polynomials

### Learning Objectives

1. Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.

### Objective 1: Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.

Stanford University psychologist and researcher, Carol Dweck, PH.D., published a book in 2006 called "Mindset, The New Psychology of Success", which changed how many people think about their talents and abilities. Based on decades of research Dr. Dweck outlined two mindsets and their influence on our learning.

Dr. Dweck’s research found that people who believe that their abilities could change through learning and practice (growth mindset) more readily accepted learning challenges and persisted through these challenges.
While individuals who believe that knowledge and abilities come from natural talent and cannot be changed (fixed mindset) more often become discouraged by failure and do not persist.
Her research shows that if we believe we can learn and master something new, this belief greatly improves our ability to learn.

1.
Read through the following illustration based on Dr. Dweck’s work.

2.
It’s important to note that we as individuals do not have a strict fixed or growth mindset at all times. We can lean one way or another in certain situations or when working in different disciplines or areas. For example, a person who often plays video games may feel they can learn any new game that is released and be confident in these abilities, but at the same time avoid sports and are fixed on the idea that they will never excel at physical activities.

In terms of learning new skills in mathematics, which mindset, growth or fixed, best describes your beliefs as of today? Explain.

3.
Identify each of the following statements as coming from a student with a fixed mindset or with a growth mindset.

4.
Mindsets can be changed. As Dr. Dweck would say “You have a choice. Mindsets are just beliefs. They are powerful beliefs, but they are something in your mind and you can change your mind.”
Think about what you would like to achieve in your classes this term and how a growth mindset can help you reach these goals. Write three goals for yourself below.

Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in , we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.
First find the area of the square in square feet.
Then find the area of the triangle in square feet.
Next find the area of the rectangular door in square feet.
The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get
  or
  ft2.
In this section, we will examine expressions such as this one, which combine several variable terms.

### Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as
  is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product
  such as
  is a term of a polynomial. If a term does not contain a variable, it is called a constant.
A polynomial containing only one term, such as
  is called a monomial. A polynomial containing two terms, such as
  is called a binomial. A polynomial containing three terms, such as
  is called a trinomial.
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

### Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,
  and
  are like terms, and can be added to get
  but
  and
  are not like terms, and therefore cannot be added.

### Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

### Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the
  in
  to obtain the equivalent expression
  When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

### Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.
The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

### Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

### Difference of Squares

Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply
  using the FOIL method.
The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.
Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

### Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

### Key Equations

### Key Concepts

1. A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See .
2. We can add and subtract polynomials by combining like terms. See  and .
3. To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See .
4. FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See .
5. Perfect square trinomials and difference of squares are special products. See  and .
6.  Follow the same rules to work with polynomials containing several variables. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, identify the degree of the polynomial.
For the following exercises, find the sum or difference.
For the following exercises, find the product.
For the following exercises, expand the binomial.
For the following exercises, multiply the binomials.
For the following exercises, multiply the polynomials.

### Real-World Applications

### Extensions

For the following exercises, perform the given operations.


# Prerequisites

## Factoring Polynomials

### Learning Objectives:

1. Master a proven technique for note taking: The Cornell Method.
2. Use the Cornell Process to study from your notes.

### Objective 1: Master a proven technique for note taking: The Cornell Method.

The Cornell Method for taking notes was created by an education professor at Cornell University, Dr. Walter Pauk. The Cornell Method consists of two strategies, a format for notetaking and then the process of using your notes to study. The format for note taking involves separating out the pages of your notebook into three or four separate regions. This method can help summarize information from a lecture, a video, or a reading from your text. You don’t need to purchase Cornell paper, just divide your sheets into regions like the illustration below. Write on just one side of the paper so that you can later fold back the left column and quiz yourself using practice test questions.

The top portion is called the heading. This is where you write your name, the class, the date, the section of the text and a main topic objective for the day.

The right section is where you will take notes during class. Try to summarize main ideas here without copying word for word everything your teacher is saying. Use bullet points or numbers to prioritize important ideas. Include here the definitions your teacher presents, and the examples worked in class. It is useful if you can group the content here by learning objectives.
The left column about 2 inches wide is used to write questions about the main concepts that were covered in class. For example, write sample test questions over the concepts discussed in class in this column. This is also where you should write cues about the importance of the information including vocabulary terms, diagrams, formulas or the methods being used to solve. If you like to sketch this is a place to add illustrations for key ideas or icons so you can quickly identify certain components.

The summary will be written in the lower section about 2 inches from the bottom. You will complete this after class summarizing the important concepts you were taught in a short compact way. Think of this summary as what you might describe to a friend who happened to miss class that day. Reflect on the important ideas here. Add illustrations or a mind map to your summary if this helps. You will use this section to find information later when studying.

### Practice Makes Perfect

### Objective 2: Use the Cornell Process to study from your notes.

The Cornell Way. Using your Cornell notes to study is referred to as the Cornell Way.
The process includes Cornell note taking (presented in Objective 1), note making, note interacting, and note reflecting.
After taking notes in class, follow up with these study methods.

Note making: This is where you will fill in any gaps left in your Cornell notes meaning add in any missed details. Fill in your question column with sample test questions, any formulas used, and highlight, circle or star important any important ideas. Complete your summary row with a few sentences summarizing the important ideas. This should be completed right after class or within the next day. Be creative and add some color, make these notes into something you enjoy working with.

Note interacting: This is the ongoing process of studying from your notes. Fold your left question column back and ask yourself the practice test questions. Include note interactions in the review for your exams. These note interactions can be as short as 5 minutes in length but need to happen regularly and at least daily for the week before exams.

Note reflecting: This is where you assess how helpful your notes were. Do this right after you get back your graded exam. Did the regular note interactions help you to perform better on the exam? Were there problems similar to those you predicted on the exam? Enhance your notes with any important ideas you had initially left out. Then use what you learn from this assessment to improve your note taking in the future.

### Practice Makes Perfect

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in .
The area of the entire region can be found using the formula for the area of a rectangle.
The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of
  units2. The other rectangular region has one side of length
  and one side of length
  giving an area of
  units2. So the region that must be subtracted has an area of
  units2.
The area of the region that requires grass seed is found by subtracting
  units2. This area can also be expressed in factored form as
  units2. We can confirm that this is an equivalent expression by multiplying.
Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

### Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance,
  is the GCF of
  and
  because it is the largest number that divides evenly into both
  and
  The GCF of polynomials works the same way:
  is the GCF of
  and
  because it is the largest polynomial that divides evenly into both
  and

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

### Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial
  has a GCF of 1, but it can be written as the product of the factors
  and

Trinomials of the form
  can be factored by finding two numbers with a product of
  and a sum of
  The trinomial
  for example, can be factored using the numbers
  and
  because the product of those numbers is
  and their sum is
  The trinomial can be rewritten as the product of
  and

### Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial
  can be rewritten as
  using this process. We begin by rewriting the original expression as
  and then factor each portion of the expression to obtain
  We then pull out the GCF of
  to find the factored expression.

###  Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.
We can use this equation to factor any perfect square trinomial.

### Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.
We can use this equation to factor any differences of squares.

### Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.
Similarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.
We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.
The sign of the first 2 is the same as the sign between
  The sign of the
  term is opposite the sign between
  And the sign of the last term, 4, is always positive.

### Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance,
  can be factored by pulling out
  and being rewritten as

### Key Equations

1. The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See .
2. Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See .
3. Trinomials can be factored using a process called factoring by grouping. See .
4. Perfect square trinomials and the difference of squares are special products and can be factored using equations. See  and .
5. The sum of cubes and the difference of cubes can be factored using equations. See  and .
6. Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the greatest common factor.
For the following exercises, factor by grouping.
For the following exercises, factor the polynomial.
For the following exercises, factor the polynomials.

### Real-World Applications

For the following exercises, consider this scenario:
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of
  m2, as shown in the figure below. The length and width of the park are perfect factors of the area.
For the following exercise, consider the following scenario:
A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below.  The flagpole will take up a square plot with area
  yd2.

### Extensions

For the following exercises, factor the polynomials completely.


# Prerequisites

## Rational Expressions

### Learning Objectives

1. Identify the skills leading to successful preparation for a college level mathematics exam.

2. Create a plan for success when taking mathematics exams.

### Objective 1: Identify the skills leading to successful preparation for a college level mathematics exam.

Complete the following surveys by placing a checkmark in the a column for each strategy based on the frequency that you engaged in the strategy during your last academic term.

### Practice Makes Perfect

Practice: Identify the skills leading to successful preparation for a college level mathematics exam.

### Objective 2: Create a plan for success when taking mathematics exams.

1. It’s important to take the opportunity to reflect on your past experiences in taking math exams as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you have had in past math courses when taking an exam and list a possible solution that you could try this semester.

2. Develop your plan for success.  Keep in mind the idea of mindsets and try to approach your test taking strategies with a growth mindset. Now is the time for growth as you begin a new term. Share your plan with your study group members.

A pastry shop has fixed costs of
  per week and variable costs of
  per box of pastries. The shop’s costs per week in terms of
  the number of boxes made, is
  We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.
Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

### Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.
We can factor the numerator and denominator to rewrite the expression.
Then we can simplify that expression by canceling the common factor

### Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

### Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite
  as the product
  Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

### Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were
  and
  then the LCD would be

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of
  by
  and the expression with a denominator of
  by

### Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression
  can be simplified by rewriting the numerator as the fraction
  and combining the expressions in the denominator as
  We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get
  which is equal to

### Key Concepts

1. Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See .
2. We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See .
3. To divide rational expressions, multiply by the reciprocal of the second expression. See .
4. Adding or subtracting rational expressions requires finding a common denominator. See  and .
5. Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, simplify the rational expressions.
For the following exercises, multiply the rational expressions and express the product in simplest form.
For the following exercises, divide the rational expressions.
For the following exercises, add and subtract the rational expressions, and then simplify.
For the following exercises, simplify the rational expression.

### Real-World Applications

### Extensions

For the following exercises, perform the given operations and simplify.

### Chapter Review Exercises

### Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.
For the following exercises, solve the equation.
For the following exercises, simplify the expression.
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

### Exponents and Scientific Notation

For the following exercises, simplify the expression.

### Radicals and Rational Expressions

For the following exercises, find the principal square root.

### Polynomials

For the following exercises, perform the given operations and simplify.

### Factoring Polynomials

For the following exercises, find the greatest common factor.
For the following exercises, factor the polynomial.

### Rational Expressions

For the following exercises, simplify the expression.

### Chapter Practice Test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
For the following exercises, evaluate the expression.
For the following exercises, simplify the expression.
For the following exercises, factor the polynomial.
For the following exercises, simplify the expression.


# Equations and Inequalities

## Introduction to Equations and Inequalities

Irrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and productive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related equipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation systems and conduits can be installed within the plotted area.


# Equations and Inequalities

## The Rectangular Coordinate Systems and Graphs

### Learning Objectives

1. Plot points on a real number line (IA 1.4.7)
2. Plot points in a rectangular coordinate system (IA 3.1.1)

### Objective 1:  Plot points on a real number line (IA 1.4.7)

### Practice Makes Perfect

Plot each set of points on a real number line.

### Objective 2: Plot points in a rectangular coordinate system (IA 3.1.1)

### Practice Makes Perfect

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in . Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.

### Plotting Ordered Pairs in the Cartesian Coordinate System

An old story describes how seventeenth-century philosopher/mathematician René Descartes, while sick in bed, invented the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.
While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the  and the vertical axis the .
The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in
The center of the plane is the point at which the two axes cross. It is known as the origin, or point
  From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis; decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative infinity as shown by the arrowheads in .
Each point in the plane is identified by its , or horizontal displacement from the origin, and its , or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form
  An ordered pair is also known as a coordinate pair because it consists of x- and y-coordinates. For example, we can represent the point
  in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See .
When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.

### Graphing Equations by Plotting Points

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.
Suppose we want to graph the equation
  We can begin by substituting a value for x into the equation and determining the resulting value of y. Each pair of x- and y-values is an ordered pair that can be plotted.  lists values of x from –3 to 3 and the resulting values for y.
We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See . This is not true for all equations.
Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

### Graphing Equations with a Graphing Utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style
  The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.
For example, the equation
  has been entered in the TI-84 Plus shown in a. In b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows
  and
  See c.
By changing the window to show more of the positive x-axis and more of the negative y-axis, we have a much better view of the graph and the x- and y-intercepts. See a and b.

### Finding x-intercepts and y-intercepts

The intercepts of a graph are points at which the graph crosses the axes. The  is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The  is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero.
To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation

To find the x-intercept, set

To find the y-intercept, set

We can confirm that our results make sense by observing a graph of the equation as in . Notice that the graph crosses the axes where we predicted it would.

### Using the Distance Formula

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,
  is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse. See .
The relationship of sides
  and
  to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,
  ) The symbols
  and
  indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.
It follows that the distance formula is given as
We do not have to use the absolute value symbols in this definition because any number squared is positive.

### Using the Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment,
  and
  the midpoint formula states how to find the coordinates of the midpoint

A graphical view of a midpoint is shown in . Notice that the line segments on either side of the midpoint are congruent.

### Key Concepts

1. We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis. See .
2. An equation can be graphed in the plane by creating a table of values and plotting points. See .
3. Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y=_____. See .
4. Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the axes. See .
5. The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See  and .
6. The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See  and .

### Section Exercises

### Verbal

### Algebraic

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.
For each of the following exercises, solve the equation for y in terms of x.
For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

### Graphical

For each of the following exercises, identify the information requested.
For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
For each of the following exercises, construct a table and graph the equation by plotting at least three points.

### Numeric

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.
For each of the following exercises, use the graph in the figure below.

### Technology

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.
For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.
Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

### Extensions

### Real-World Applications


# Equations and Inequalities

## Linear Equations in One Variable

### Learning Objectives

1. Simplify expressions using order of operations (IA 1.1.3)
2. Solve linear equations using a general strategy (IA 2.1.1)

### Objective 1: Simplify expressions using order of operations (IA 1.1.3)

### Practice Makes Perfect

Evaluate the following expressions being sure to follow the order of operations:
Simplify by combining like terms:

### Objective 2: Solve linear equations using a general strategy (IA 2.1.1)

### Practice Makes Perfect

Solve each linear equation using the general strategy.
Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in .

### Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form
  and are solved using basic algebraic operations.
We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.
The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for
  will make the equation true.
A conditional equation is true for only some values of the variable. For example, if we are to solve the equation
  we have the following:
The solution set consists of one number:
  It is the only solution and, therefore, we have solved a conditional equation.
An inconsistent equation results in a false statement. For example, if we are to solve
  we have the following:
Indeed,
  There is no solution because this is an inconsistent equation.
Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

### Solving a Rational Equation

In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a rational equation.
Recall that a rational number is the ratio of two numbers, such as
  or
  A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.
Rational equations have a variable in the denominator in at least one of the terms.
Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).
Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.
A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as
  Always consider a binomial as an individual factor—the terms cannot be separated. For example, suppose a problem has three terms and the denominators are

  and
  First, factor all denominators. We then have

  and
  as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of
  The
  in the first denominator is separate from the
  in the
  denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the
  one factor of
  and the 3. Thus, the LCD is the following:
So, both sides of the equation would be multiplied by
  Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.
Another example is a problem with two denominators, such as
  and
  Once the second denominator is factored as
  there is a common factor of x in both denominators and the LCD is

Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.
We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.
Multiply
  and
  which results in

Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.

### Finding a Linear Equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as
  where
  and
  Let us begin with the slope.

### The Slope of a Line

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in . The lines indicate the following slopes:

  and

### The Point-Slope Formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

### Standard Form of a Line

Another way that we can represent the equation of a line is in standard form. Standard form is given as
where

  and
  are integers. The x- and y-terms are on one side of the equal sign and the constant term is on the other side.

### Vertical and Horizontal Lines

The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as
where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.
Suppose that we want to find the equation of a line containing the following points:
  and
  First, we will find the slope.
Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through
  See .
The equation of a horizontal line is given as
where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.
Suppose we want to find the equation of a line that contains the following set of points:
  and
  We can use the point-slope formula. First, we find the slope using any two points on the line.
Use any point for
  in the formula, or use the y-intercept.
The graph is a horizontal line through
  Notice that all of the y-coordinates are the same. See .

### Determining Whether Graphs of Lines are Parallel or Perpendicular

Parallel lines have the same slope and different y-intercepts. Lines that are parallel to each other will never intersect. For example,  shows the graphs of various lines with the same slope,

All of the lines shown in the graph are parallel because they have the same slope and different y-intercepts.
Lines that are perpendicular intersect to form a
  -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is
  For example,  shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of

### Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

### Key Concepts

1. We can solve linear equations in one variable in the form
  using standard algebraic properties. See  and .
2. A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See  and .
3. All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See  and  and .
4. Given two points, we can find the slope of a line using the slope formula. See .
5. We can identify the slope and y-intercept of an equation in slope-intercept form. See .
6. We can find the equation of a line given the slope and a point. See .
7. We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See .
8. The standard form of a line has no fractions. See .
9. Horizontal lines have a slope of zero and are defined as
  where c is a constant.
10. Vertical lines have an undefined slope (zero in the denominator), and are defined as
  where c is a constant. See .
11. Parallel lines have the same slope and different y-intercepts. See  and .
12. Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the equation for

For the following exercises, solve each rational equation for
  State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.
Write all the final equations using the slope-intercept form.
For the following exercises, find the equation of the line using the given information.

### Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

### Numeric

For the following exercises, find the slope of the line that passes through the given points.
For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

### Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

### Extensions

### Real-World Applications

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be
  where
  is the number of miles traveled.


# Equations and Inequalities

## Models and Applications

### Learning Objectives

1. Solve a formula for a specified variable (IA 2.3.1)
2. Use a problem-solving strategy for word problems (IA 2.2.1)

### Objective 1: Solve a formula for a specified variable (IA 2.3.1)

### Practice Makes Perfect

Solve each formula for the specific variable.

### Objective 2: Use a problem-solving strategy for word problems (IA 2.2.1)

### Practice Makes Perfect

Use a Problem-Solving Strategy for word problems.
Neka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum number of points that can be earned is 100. Is it possible for Neka to end the course with an A? A simple linear equation will give Neka his answer.
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

### Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write
  This expression represents a variable cost because it changes according to the number of miles driven.
If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost

When dealing with real-world applications, there are certain expressions that we can translate directly into math.  lists some common verbal expressions and their equivalent mathematical expressions.

### Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region,
  the perimeter of a rectangle,
  and the volume of a rectangular solid,
  When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Key Concepts

1. A linear equation can be used to solve for an unknown in a number problem. See .
2. Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See .
3. There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the
  formula. See .
4. Many geometry problems are solved using the perimeter formula
  the area formula
  or the volume formula
  See , , and .

### Section Exercises

### Verbal

### Real-World Applications

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.
For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.
For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use
  for the number of devices that need data plans as part of their cost.
For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.
For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.
For the following exercises, use the formula given to solve for the required value.
For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.