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

17.
19.
20.
x 2 2x 3 < 0 18.
x 3x 1 < 0
Critical numbers: x 3, x 1
Test intervals: , 3, 3, 1, 1,
Test: Is x 3x 1 < 0?
Interval x-Value Value of
x 3x 1
Conclusion
, 3
3, 1
1,
Solution set:
x 4
x 0
x 2
3, 1
x 2 8x 5 ≥ 0
x 2 8x 5 0
x 2 8x 16 5 16
x 4 2 21
x 4 ±21
15 5
31 3
51 5
−3
x 4 ± 21
−2
−1
0
1
Positive
Negative
Positive
Complete the square.
x
x
Section 2.7 Nonlinear Inequalities 225
x 2 4x 1 > 0
4 ± 16 4
2
Critical numbers:
Test intervals:
Solution intervals:
2 − 5 2 + 5
−4 −2 0 2 4 6 8
Critical numbers:
Test intervals:
Test: Is x2 x 4 ± 21
, 4 21, 4 21, 4 21, 4 21,
8x 5 ≥ 0?
Interval x-Value Value of x Conclusion
, 4 21 x 10 100 80 5 15 Positive
4 21, 4 21 x 0 0 0 5 5 Negative
4 21,
x 2 4 16 5 15 Positive
2 8x 5
2 ± 5
x 2 5, x 2 5
, 2 5 ⇒ x 2 4x 1 > 0
2 5, 2 5 ⇒ x 2 4x 1 < 0
2 5, ⇒ x
, 2 5 2 5,
2 4x 1 > 0
−4 − 21
Solution set: < 4 21 4 21, −10 −8 −6 −4 −2 0 2
2x 2 6x 15 ≤ 0
2 x 2 6x 15 ≥ 0
x 6 ± 62 4215
22
Critical numbers:
Test intervals:
3
2
Solution interval:
x 3
2
3 39
2 2 , ⇒ 2x2 6x 15 < 0
3 39
,
2 2
6 ± 156
4
39 3 39
, x
2 2 2
3 39
,
2 2 ⇒ 2x2 6x 15 < 0
39 3 39
,
2 2 2 ⇒ 2x2 6x 15 > 0
3 39
2 2 ,
6 ± 239
4
3 39
−
2 2
−2 −1
3 39
±
2 2
0
1
2
3
3 39
+
2 2
4
5
x
x
−4 + 21
x