C H A P T E R 2 Polynomial and Rational Functions

226 Chapter 2 Polynomial and Rational Functions
21.
23.
24.
Critical numbers: x ±1, x 3
Test intervals: , 1, 1, 1, 1, 3, 3,
Test: Is x 1x 1x 3 > 0?
Interval x-Value Value of
x 1x 1x 3
Conclusion
1, 1
1, 3
3,
x 3 3x 2 x 3 > 0 22.
x 2 x 3 1x 3 > 0
, 1
Solution set:
x 3 2x 2 9x 2 ≥ 20
x 3 2x 2 9x 18 ≥ 0
x 2 x 2 9x 2 ≥ 0
x 2x 2 9 ≥ 0
x 2x 3x 3 ≥ 0
Critical numbers: x 2, x ±3
Test intervals: , 3, 3, 2, 2, 3, 3,
Test: Is x 2x 3x 3 ≥ 0?
Interval x-Value Value of x 2x 3x 3 Conclusion
, 3 x 4 617 42 Negative
3, 2 x 0 233 18
Positive
2, 3 x 2.5 0.55.50.5 1.375 Negative
3, x 4 271 14
Positive
Solution set: 3, 2 3,
2x 3 13x 2 8x 46 ≥ 6
2x 3 13x 2 8x 52 ≥ 0
x 2 2x 13 42x 13 ≥ 0
Critical numbers:
Test intervals:
x 2 1x 3 > 0
x 1x 1x 3 > 0
x 2
x 0
x 2
x 4
1, 1 3,
2x 13x 2 4 ≥ 0
113 3
311 3
531 15
x 13
2 , x 2, x 2
, 13
2 ⇒ 2x 3 13x 2 8x 52 < 0
13
2 , 2 ⇒ 2x3 13x 2 8x 52 > 0
2, 2 ⇒ 2x 3 13x 2 8x 52 < 0
2, ⇒ 2x 3 13x 2 8x 52 > 0
Solution interval: 13
2 , 2, 2,
135 15
−1
Negative
Positive
Negative
Positive
0 1 2 3 4
x
13
−
2
x 3 2x 2 4x 8 ≤ 0
x 2 x 2 4x 2 ≤ 0
x 2x 2 4 ≤ 0
Critical numbers:
Test intervals: , 2 ⇒ x3 2x2 4x 8 < 0
Solution interval: , 2
0 1 2 3 4
−4 −3 −2 −1 0 1 2 3 4 5
−8 −6 −4 −2 0 2
2, 2 ⇒ x 3 2x 2 4x 8 < 0
2, ⇒ x 3 2x 2 4x 8 > 0
4
x 2, x 2
x
x
x