2 and Problem Solving Equations, Inequalities,

2 CHAPTER

Equations, Inequalities, and Problem Solving

2.1 Linear Equations in One

Variable

2.2 An Introduction to

Problem Solving

2.3 Formulas and Problem

Solving

2.4 Linear Inequalities and

Problem Solving

Integrated Review-- Linear Equations and Inequalities

Today, it seems that most people in the world want to stay connected most of the time. In fact, 86% of U.S. citizens own cell phones. Also, computers with Internet access are just as important in our lives. Thus, the merging of these two into Wi-Fi-enabled cell phones might be the next big technological explosion. In Section 2.1, Objective 1, and Section 2.2, Exercises 35 and 36, you will find the projected increase in the number of Wi-Fi-enabled cell phones in the United States as well as the percent increase. (Source: )

2.5 Compound Inequalities

2.6 Absolute Value

Equations

2.7 Absolute Value

Inequalities

Number of Wi-Fi-Enabled Cell Phones in the U.S. (in millions)

Projected Growth of Wi-Fi-Enabled Cell Phones in the U.S.

160 150 140 130 120 110 100 90 80 70 60 50 40 30

2009 2010 2011 2012 2013 2014 2015

Year

Mathematics is a tool for solving problems in such diverse fields as transportation, engineering, economics, medicine, business, and biology. We solve problems using mathematics by modeling real-world phenomena with mathematical equations or inequalities. Our ability to solve problems using mathematics, then, depends in part on our ability to solve equations and inequalities. In this chapter, we solve linear equations and inequalities in one variable and graph their solutions on number lines.

47

48 CHAPTER 2 Equations, Inequalities, and Problem Solving

2.1 Linear Equations in One Variable

OBJECTIVES

1 Solve Linear Equations Using

Properties of Equality.

2 Solve Linear Equations

That Can Be Simplified by Combining Like Terms.

3 Solve Linear Equations

Containing Fractions or Decimals.

4 Recognize Identities and

Equations with No Solution.

OBJECTIVE

1 Solving Linear Equations Using Properties of Equality

Linear equations model many real-life problems. For example, we can use a linear equation to calculate the increase in the number (in millions) of Wi-Fi-enabled cell phones.

Wi-Fi-enabled cell phones let you carry your Internet access with you. There are already several of these smart phones available, and this technology will continue to expand. Predicted numbers of Wi-Fi-enabled cell phones in the United States for various years are shown below.

Number of Wi-Fi-Enabled Cell Phones in the U.S. (in millions)

Projected Growth of Wi-Fi-Enabled Cell Phones in the U.S.

160

150

149

140

138

130

120

118

110

100

97

90

80

76

70

60

55

50

40

38

30

0

2009

2010

2011

2012

2013

2014

2015

Year

To find the projected increase in the number of Wi-Fi-enabled cell phones in the United States from 2014 to 2015, for example, we can use the equation below.

Increase in

cell phones in

cell phones in

In words: cell phones is

2015

minus

2014

Translate:

x

=

149

-

138

Since our variable x (increase in Wi-Fi-enabled cell phones) is by itself on one side of the equation, we can find the value of x by simplifying the right side.

x = 11

The projected increase in the number of Wi-Fi-enabled cell phones from 2014 to 2015 is 11 million.

The equation x = 149 - 138, like every other equation, is a statement that two expressions are equal. Oftentimes, the unknown variable is not by itself on one side of the equation. In these cases, we will use properties of equality to write equivalent equations so that a solution may be found. This is called solving the equation. In this section, we concentrate on solving equations such as this one, called linear equations in one variable. Linear equations are also called first-degree equations since the exponent on the variable is 1.

3x = -15

Linear Equations in One Variable 7 - y = 3y 4n - 9n + 6 = 0 z = -2

Section 2.1 Linear Equations in One Variable 49

Linear Equations in One Variable

A linear equation in one variable is an equation that can be written in the form ax + b = c

where a, b, and c are real numbers and a 0.

When a variable in an equation is replaced by a number and the resulting equation is true, then that number is called a solution of the equation. For example, 1 is a solution of the equation 3x + 4 = 7, since 3112 + 4 = 7 is a true statement. But 2 is not a solution of this equation, since 3122 + 4 = 7 is not a true statement. The solution set of an equation is the set of solutions of the equation. For example, the solution set of 3x + 4 = 7 is 516.

To solve an equation is to find the solution set of an equation. Equations with the same solution set are called equivalent equations. For example,

3x + 4 = 7 3x = 3 x = 1

are equivalent equations because they all have the same solution set, namely 516. To solve an equation in x, we start with the given equation and write a series of simpler equivalent equations until we obtain an equation of the form

x number

Two important properties are used to write equivalent equations.

The Addition and Multiplication Properties of Equality

If a, b, and c, are real numbers, then a = b and a + c = b + c are equivalent equations.

Also, a = b and ac = bc are equivalent equations as long as c 0.

The addition property of equality guarantees that the same number may be added to

both sides of an equation, and the result is an equivalent equation. The multiplication

property of equality guarantees that both sides of an equation may be multiplied by

the same nonzero number, and the result is an equivalent equation. Because we define

subtraction in terms of addition 1a - b = a + 1 -b2 2, and division in terms of multi-

plication a a = a # 1 b , these properties also guarantee that we may subtract the same

b

b

number from both sides of an equation, or divide both sides of an equation by the

same nonzero number and the result is an equivalent equation. For example, to solve 2x + 5 = 9, use the addition and multiplication properties

of equality to isolate x--that is, to write an equivalent equation of the form

x number

We will do this in the next example.

E X A M P L E 1 Solve for x: 2x + 5 = 9.

Solution First, use the addition property of equality and subtract 5 from both sides. We do this so that our only variable term, 2x, is by itself on one side of the equation.

2x + 5 = 9 2x + 5 - 5 = 9 - 5

2x = 4

Subtract 5 from both sides. Simplify.

Now that the variable term is isolated, we can finish solving for x by using the multiplication property of equality and dividing both sides by 2.

2x = 4 22 x=2

Divide both sides by 2. Simplify.

50 CHAPTER 2 Equations, Inequalities, and Problem Solving

Check: To see that 2 is the solution, replace x in the original equation with 2.

2x + 5 = 9 2122 + 5 9

4 + 59

9=9

Original equation Let x = 2.

True

Since we arrive at a true statement, 2 is the solution or the solution set is 526 .

PRACTICE

1 Solve for x: 3x + 7 = 22.

Helpful Hint Don't forget that

0.4 = c and c = 0.4 are equivalent equations.

We may solve an equation so that the variable is alone on either side of the equation.

E X A M P L E 2 Solve: 0.6 = 2 - 3.5c.

Solution We use both the addition property and the multiplication property of equality.

Check:

0.6 = 2 - 3.5c

0.6 - 2 = 2 - 3.5c - 2

-1.4 = -3.5c

- 1.4 - 3.5

=

- 3.5c - 3.5

0.4 = c

Subtract 2 from both sides. Simplify. The variable term is now isolated.

Divide both sides by - 3.5.

- 1.4

Simplify

-

. 3.5

0.6 = 2 - 3.5c 0.6 2 - 3.510.42 0.6 2 - 1.4

0.6 = 0.6

Replace c with 0.4. Multiply. True

The solution is 0.4, or the solution set is 50.46 .

PRACTICE

2 Solve: 2.5 = 3 - 2.5t.

OBJECTIVE

2 Solving Linear Equations That Can Be Simplified by Combining Like Terms

Often, an equation can be simplified by removing any grouping symbols and combining any like terms.

E X A M P L E 3 Solve: -4x - 1 + 5x = 9x + 3 - 7x.

Solution First we simplify both sides of this equation by combining like terms. Then, let's get variable terms on the same side of the equation by using the addition property of equality to subtract 2x from both sides. Next, we use this same property to add 1 to both sides of the equation.

-4x - 1 + 5x = 9x + 3 - 7x x - 1 = 2x + 3

x - 1 - 2x = 2x + 3 - 2x -x - 1 = 3

-x - 1 + 1 = 3 + 1 -x = 4

Combine like terms. Subtract 2x from both sides. Simplify. Add 1 to both sides. Simplify.

Notice that this equation is not solved for x since we have - x or - 1x, not x. To solve for x, we divide both sides by - 1.

-x -1

=

4 -1

Divide both sides by - 1.

x = - 4 Simplify.

Section 2.1 Linear Equations in One Variable 51

Check to see that the solution is -4, or the solution set is 5 -46.

PRACTICE

3 Solve: -8x - 4 + 6x = 5x + 11 - 4x.

If an equation contains parentheses, use the distributive property to remove them.

E X A M P L E 4 Solve: 21x - 32 = 5x - 9. Solution First, use the distributive property.

2(x-3)=5x-9 2x - 6 = 5x - 9 Use the distributive property.

Next, get variable terms on the same side of the equation by subtracting 5x from both sides.

2x - 6 - 5x = 5x - 9 - 5x

-3x - 6 = -9

-3x - 6 + 6 = -9 + 6

-3x = -3

- 3x -3

=

-3 -3

x=1

Subtract 5x from both sides. Simplify. Add 6 to both sides. Simplify.

Divide both sides by - 3.

Let x = 1 in the original equation to see that 1 is the solution.

PRACTICE

4 Solve: 31x - 52 = 6x - 3.

OBJECTIVE

3 Solving Linear Equations Containing Fractions or Decimals

If an equation contains fractions, we first clear the equation of fractions by multiplying both sides of the equation by the least common denominator (LCD) of all fractions in the equation.

EXAMPLE 5

Solve for y:

y

-

y

=

1 .

34 6

Solution First, clear the equation of fractions by multiplying both sides of the equation by 12, the LCD of denominators 3, 4, and 6.

y-y=1 34 6

y 12 a

-

y b

=

12a 1 b

Multiply both sides by the LCD 12.

34

6

y 12a b

-

y 12a b

=

2

3

4

4y - 3y = 2

Apply the distributive property. Simplify.

y=2

Simplify.

Check: To check, let y = 2 in the original equation.

y-y=1 34 6 2 - 21 346

Original equation. Let y = 2.

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