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

Section 2.7 Nonlinear Inequalities 231
47.
xx 2
≤ 0
x 3x 3
Critical numbers: x 0, x 2, x ±3
Test intervals:
, 3, 3, 2, 2, 0, 0, 3, 3,
xx 2
Test: Is
≤ 0?
x 3x 3
By testing an x-value in each test interval in the inequality,
we see that the solution set is: 3, 2 0, 3
x2 2x
x2 ≤ 0
9
48.
x 3x 2
≥ 0
x
Critical numbers: x 3, x 0, x 2
x 3x 2
Test intervals: , 3 ⇒ < 0
x
x 3x 2
3, 0 ⇒ > 0
x
x 3x 2
0, 2 ⇒ < 0
x
x2 x 6
≥ 0
x
49.
51.
−3 −2 −1 0 1 2 3
5x 1 2xx 1 x 1x 1
< 0
x 1x 1
Critical numbers:
Test intervals:
x
Test: Is
By testing an x-value in each test interval in the inequality,
we see that the solution set is: , 1 2
3 , 1 3, 3x 2x 3
< 0?
x 1x 1
2
−
3
−1
0
1
y 3x
x 2
8
5x 5 2x2 2x x 2 1
x 1x 1
−6 12
−4
2
, 1, 1, 2 3 , 2 3 , 1 , 1, 3, 3,
3
x 2
, x 3, x ±1
3
4
5 2x
< 1 50.
x 1 x 1
5 2x
1 < 0
x 1 x 1
3x2 7x 6
x 1x 1
3x 2x 3
< 0
x 1x 1
x
< 0
< 0
(a) y ≤ 0 when 0 ≤ x < 2.
(b) y ≥ 6 when 2 < x ≤ 4.
2, ⇒
Solution intervals: 3, 0 2,
−3 −2 −1 0 1 2 3
Critical numbers:
Test intervals: , 4 ⇒
x
x 3x 2
x
> 0
Solution intervals: , 4, 2, 1, 6,
−4
−2
1
4, 2 ⇒
2, 1 ⇒
1, 6 ⇒
6, ⇒
0 2 4 6
x
3x
x 1 ≤
3xx 4 xx 1 3x 4x 1
≤ 0
x 1x 4
x2 4x 12
x 1x 4
x 6x 2
≤ 0
x 1x 4
x 4, x 2, x 1, x 6
x 6x 2
x 1x 4
x 6x 2
x 1x 4
x 6x 2
x 1x 4
x 6x 2
x 1x 4
≤ 0
x 6x 2
x 1x 4
x
3
x 4
> 0
< 0
< 0
< 0
> 0