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

25.
27.
29.
31.
4x 2 4x 1 ≤ 0 26.
Critical number:
Test intervals:
Test: Is
Interval x-Value Value of 2x 1 Conclusion
2
1
2 ,
2x 1 2 ≤ 0
1
, 2
Solution set:
x 0
x 1
x 1
2
x 1
2
1
, 2 , 1
2 ,
2x 1 2 ≤ 0?
−2
1 2 1
1 2 1
−1
0
1
2
1
2
Positive
Positive
4x 3 6x 2 < 0 28.
2x 2 2x 3 < 0
Critical numbers:
Test intervals:
Test: Is
2x2 , 0, 0,
2x 3 < 0?
3
2, 3 2 ,
By testing an x-value in each test interval in the inequality,
we see that the solution set is: , 0 0, 3
2
Critical numbers:
Test intervals:
x 0, x 3
2
x 3 4x ≥ 0 30.
xx 2x 2 ≥ 0
x 0, x ±2
, 2, 2, 0, 0, 2, 2,
Test: Is xx 2x 2 ≥ 0?
By testing an x-value in each test interval in the inequality,
we see that the solution set is: 2, 0 2,
x 1 2 x 2 3 ≥ 0 32.
Critical numbers:
Test intervals: , 2, 2, 1, 1, )
Test: Is
x 1, x 2
x 1 2 x 3 3 ≥ 0?
By testing an x-value in each test interval in the inequality,
we see that the solution set is: 2,
x
Section 2.7 Nonlinear Inequalities 227
x 2 3x 8 > 0
The critical numbers are imaginary:
3 i23
±
2 2
So the set of real numbers is the solution set.
−3 −2 −1
4x 3 12x 2 > 0
4x 2 x 3 > 0
Critical numbers:
0
1 2 3
Test intervals: , 0 ⇒ 4x2x 3 < 0
Solution interval: 3,
2x 3 x 4 ≤ 0
x 3 2 x ≤ 0
Critical numbers:
x
x 0, x 3
0, 3 ⇒ 4x 2 x 3 < 0
3, ⇒ 4x 2 x 3 > 0
x 0, x 2
Test intervals: , 0 ⇒ x32 x < 0
0, 2 ⇒ x 3 2 x > 0
2, ⇒ x 3 2 x < 0
Solution intervals: , 0 2,
x 4 x 3 ≤ 0
Critical numbers:
x 0, x 3
Test intervals: , 0 ⇒ x4x 3 < 0
0, 3 ⇒ x 4 x 3 < 0
3, ⇒ x 4 x 3 > 0
Solution intervals: , 0 0, 3 or , 3