Intermediate Algebra (11th edition)

104 CHAPTER 2 Linear Equations, Inequalities, and Applications

Solving a Compound Inequality with and

Solve the compound inequality, and graph the solution set.

and

Step 1 Solve each inequality individually.

and

and

and

Step 2 Because of the word and,the solution set will include all numbers that satisfy

both inequalities in Step 1 at the same time. The compound inequality is true

whenever x... 8 and xÚ 5 are both true. See the graphs in FIGURE 17.

x... 8 xÚ 5

x+ 1 - 1 ... 9 - 1 x - 2 + 2 Ú 3 + 2

x+ 1 ... 9 x - 2 Ú 3

x+ 1 ... 9 x- 2 Ú 3

EXAMPLE 2

Solving a Compound Inequality with and

Solve the compound inequality, and graph the solution set.

and

Step 1 Solve each inequality individually.

and

and

and

The graphs of x6-and x...- 4 are shown in FIGURE 19.

7

3

x 6 - x...- 4

7

3

- 3 x 77 5 x...- 20

- 3 x- 275 5 x- 1 ...- 21

- 3 x- 275 5 x- 1 ...- 21

EXAMPLE 3

The set of points

where the graphs

“overlap” represents

the intersection.

NOW TRY
EXERCISE 2
Solve the compound
inequality, and graph the
solution set.

x- 2 ... 5 and x+ 5 Ú 9

The intersection of the two graphs is the solution set of the compound

inequality. FIGURE 18shows that the solution set, in interval notation, is 3 5, 8 4.

NOW TRY

Step 2 Now find all values of xthat are less than and also less than or equal to

- 4.As shown in FIGURE 20, the solution set is 1 - q, - 44.

-

7

3

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

x ≤ 8

x ≥ 5

FIGURE 17

091 2 3 4 5 6 7 8

[5, 8]

FIGURE 18

Remember to

reverse the

inequality symbol.

–5 –4 –3 –2 –1 0

–5 –4 –3 –2 –1 0

x < –

x ≤ –4

7

• 3

7

3

FIGURE 19

–5 –4 –3 –2 –1 0

(–∞, –4]

FIGURE 20 NOW TRY

NOW TRY
EXERCISE 3
Solve and graph.

and

3 x+ 4 Ú- 5

• 4 x- 167

NOW TRY ANSWERS
2.

3. 1 - 2, q 2

3 4, 7 4

345678

–3 –2 –1 0 1