Intermediate Algebra (11th edition)

112 CHAPTER 2 Linear Equations, Inequalities, and Applications

OBJECTIVES Suppose that the government of a country decides that it will comply with a certain

restriction on greenhouse gas emissions within3 years of 2020. This means that the

differencebetween the year it will comply and 2020 is less than 3, without regard to

sign. We state this mathematically as

Absolute value inequality

where xrepresents the year in which it complies.

Reasoning tells us that the year must be between 2017 and 2023, and thus

makes this inequality true. But what general procedure is used to

solve such an inequality? We now investigate how to solve absolute value equations

and inequalities.

OBJECTIVE 1 Use the distance definition of absolute value.In Section 1.1,

we saw that the absolute value of a number x, written represents the distance

from xto 0 on the number line. For example, the solutions of are 4 and ,

as shown in FIGURE 29.

|x|= 4 - 4

|x|,

20176 x 62023

|x- 2020 | 6 3,

Absolute Value Equations and Inequalities

2.7

1 Use the distance

definition of

absolute value.

2 Solve equations

of the form

for

3 Solve inequalities

of the form

and

of the form

for

4 Solve absolute value

equations that

involve rewriting.

5 Solve equations

of the form

6 Solve special cases

of absolute value

equations and

inequalities.

|ax+b|=|cx+d|.

k 7 0.

|ax+b| 7 k,

|ax+b| 6 k

k 7 0.

|ax+b|=k,

–4 0

x = – 4 or x = 4

4 units from 0 4 units from 0

4

FIGURE 29

–4 04

More than

4 units from 0

More than

4 units from 0

x < – 4 or x > 4

FIGURE 30

Less than 4 units from 0

–4 04

• 4 < x < 4
  FIGURE 31

Because absolute value represents distance from 0, we interpret the solutions

of to be all numbers that are morethan four units from 0. The set

fits this description. FIGURE 30shows the graph of the solution set

of Because the graph consists of two separate intervals, the solution set is

described using the word or:or.x6- 4 x 74

|x| 7 4.

1 - q, - 42 ́ 1 4, q 2

|x| 7 4

The solution set of consists of all numbers that are lessthan 4 units from

0 on the number line. This is represented by all numbers between and 4. This set

of numbers is given by as shown in FIGURE 31. Here, the graph shows that

- 46 x 6 4,which means x7- 4 and x 6 4.

1 - 4, 4 2 ,

- 4

|x| 64