Beginning Algebra, 11th Edition

Graphing a Three-Part Inequality

Write the inequality in interval notation, and graph the interval.

The statement is read “ is less than or equal to x and xis less than 3.” We want

the set of numbers between and 3, with included and 3 excluded. In interval

notation, we write using a square bracket at because is part of the

graph and a parenthesis at 3 because 3 is not part of the graph. See FIGURE 26.

3 - 1, 3 2 , - 1 - 1

- 1 - 1

- 1

- 1 ...x 63

EXAMPLE 7

158 CHAPTER 2 Linear Equations and Inequalities in One Variable

–3 –2 –1 0 152 3 4

–1 is included. 3 is excluded.

FIGURE 26Graph of the interval 3 - 1, 3 2

NOW TRY

The three-part inequality

says that is between 3 and 8. We solve this inequality as follows.

Subtract 2 from each part.

The idea is to get the inequality in the form

a number 6 x 6 another number.

16 x 66

3 - 26 x+ 2 - 268 - 2

x+ 2

36 x+ 26 8

CAUTION Three-part inequalities are written so that the symbols point in

the same direction and both point toward the lesser number.It would be wrongto

write an inequality as since this would imply that a false

statement.

86 x+ 26 3, 86 3,

Solving Three-Part Inequalities

Solve each inequality, and graph the solution set.

(a)

Add 5 to each part.

Divide each part by 3.

36 x ... 5

9

3

6

3 x

3

...

15

3

96 3 x ... 15

4 + 563 x- 5 + 5 ... 10 + 5

46 3 x- 5 ... 10

EXAMPLE 8

Remember to divide

all threeparts by 3.

The solution set is 1 3, 5 4 .Its graph is shown in FIGURE 27.

–2 –1 0 1 263 4 5

3 is excluded. 5 is included.

FIGURE 27Graph of the interval 1 3, 5 4

NOW TRY
EXERCISE 7
Write the inequality in inter-
val notation, and graph the
interval.

0 ...x 62

NOW TRY ANSWER
7.

–3–2–1 0 1 2 3

3 0, 2 2

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