Section 11.5 Polynomial and Rational Inequalities
26.
2
2
3105
31050
xx
xx


Solve the related quadratic equation.
2
31050xx
Interval Test
Value Test Conclusion
510
,3



0
2
3(0) 10(0) 5

,3

27.
2
2
251
2510
xx
xx


Solve the related quadratic equation.
2
2510xx
Chapter 11 Quadratic Equations and Functions
Interval Test
Value Test Conclusion
533
,4




1
2
2( 1) 5( 1) 1
71, true
 
533
,4

belongs to the solution set.
28.
2
2
5811
58110
xx
xx


Solve the related quadratic equation.
2
58110xx
Interval Test
Value Test Conclusion
471
,5





3
2
5( 3) 8( 3) 11
21 11, true

471
,5


belongs to the solution set.
Section 11.5 Polynomial and Rational Inequalities
29.
2
690xx
Solve the related quadratic equation.
2
690
xx

The boundary point is 3.
Interval Test Value Test Conclusion
,3 0
2
06090
90, False

,3 does not belong to the solution set.
30.
2
4410xx
Solve the related quadratic equation.
2
2
4410
21 0
xx
x


Chapter 11 Quadratic Equations and Functions
31.
1230xx x 
Solve the related polynomial equation.
1230xx x 
1 0 or 2 0 or 3 0
123
xx x
xxx
  

The boundary points are 1, 2, and 3.
Interval Test Value Test Conclusion
0
,1 0 1 0 2 0 3 0 ,1 does not belong
6 0, False to the solution set.
   

32.
1230xx x 
1230xx x 
10
1
x
x


or 20
2
x
x


or 30
3
x
x


Interval Test Value Test Conclusion
, 3 4 4 1 4 2 4 3 0 false , 3 is not part of the solution set
   
Section 11.5 Polynomial and Rational Inequalities
33.
32
220xxx
Solve the related polynomial equation.
32
220
xxx

The boundary points are
2, 1, and 1
.
32
Interval Test Value Test Conclusion
3
, 2 , 2 does not belong
323 320
to the solution set.
80, False
   


34.
32
2480
xxx


32
2
2
2480
24 2 0
240
xxx
xx x
xx

 

Chapter 11 Quadratic Equations and Functions
35.
32
39270
xxx

Solve the related polynomial equation.
32
39270
xxx
 
The boundary points are
3 and 3
.
32
Interval Test Value Test Conclusion
4
, 3 , 3 belongs to the
43494270
solution set.
49 0, True
   


36.

32
32
2
2
770
770
71 7 0
710
xxx
xxx
xx x
xx


 

32
Interval Test Value Test Conclusion
, 7 8 8 7 8 8 7 0 true , 7 is part of the solution set
   
Section 11.5 Polynomial and Rational Inequalities
37.
32
440
xx x

Solve the related polynomial equation.
32
440
xx x

The imaginary solutions will not be boundary points, so the only boundary point is
1
.
32
Interval Test Value Test Conclusion
2
, 1 , 1 does not belong
224240
to the solution set.
80, False
   


38.
32
32
2
990
990
19 1 0
xx x
xx x
xx x


 
Chapter 11 Quadratic Equations and Functions
39.
32
32
9
90
xx
xx

Solve the related polynomial equation.
The boundary points are 0 and 9.
32
Interval Test Value Test Conclusion
1
,0 ,0 does not belong
191
to the solution set.
19, False
 


40.
32
32
32
2
4
4
40
40
xx
xx
xx
xx


Section 11.5 Polynomial and Rational Inequalities
41.
40
3
x
x
Find the values of x that make the numerator and denominator zero.
40 30
43
xx
xx
 

The boundary points are –3 and 4. Exclude –3 from the solution set, since this would make the denominator zero.
Interval Test Value Test Conclusion
,3 
-4
44 0
43


,3 
belongs to the solution set.
42.
50
2
x
x
Find the values of x that make the numerator and denominator zero.
50 20
52
xx
xx
 
 
Interval Test Value Test Conclusion
,5 
−6
65 0 true
62


,5 
belongs to the solution set.
Chapter 11 Quadratic Equations and Functions
43.
30
4
x
x
Find the values of x that make the numerator and denominator zero.
30 40
34
xx
xx
 
 
The boundary points are –4 and –3.
Interval Test Value Test Conclusion
,4 
−5
53 0
54
20, false


,4 
does not belong to the solution set.
44.
50
2
x
x
Find the values of x that make the numerator and denominator zero.
50 20
52
xx
xx
 
 
Interval Test Value Test Conclusion
,5 
−6
65 0 false

,5 
does not belong to the solution set.
Section 11.5 Polynomial and Rational Inequalities
45.
20
4
x
x

Find the values of x that make the numerator and denominator zero.
The boundary points are 2 and 4.
Interval Test Value Test Conclusion
,2
0
02 0
04
10, false


,2
does not belong to the solution set.
46.
30
2
x
x

Find the values of x that make the numerator and denominator zero.
30 and 20
32
3
xx
xx
x
  
 

Interval Test Value Test Conclusion
,3 
−4
43
0 true
42
 

,3 
belongs to the solution set.
Chapter 11 Quadratic Equations and Functions
47.
42 0
34
x
x
Find the values of x that make the numerator and denominator zero.
The boundary points are
4and 2
3
.
Interval Test Value Test Conclusion
4
,3

 


−2
422 0
32 4
40, true



4
,3
 
belongs to the solution set.
48.
35
0
62
x
x
Find the values of x that make the numerator and denominator zero.
and
350 62 0
35 26
xx
xx
  
  
Section 11.5 Polynomial and Rational Inequalities
49.
0
3
x
x
Find the values of x that make the numerator and denominator zero.
0and 30
3
xx
x

The boundary points are 0 and 3.
Interval Test Value Test Conclusion
,0
−1
10
13
10, true

4
,3
 
belongs to the solution set.
50.
40
x
x
Find the values of x that make the numerator and denominator zero.
40 and 0
4
xx
x
 

Interval Test Value Test Conclusion
,4 
−5
54 0 true
5

,4 
belongs to the solution set.
Chapter 11 Quadratic Equations and Functions
51.
12
3
x
x
Express the inequality so that one side is zero.
120
3
x
x

The boundary points are –5 and –3.
Interval Test Value Test Conclusion
,5 
−6
61 2
63
52, true
3


,5 
belongs to the solution set.
Section 11.5 Polynomial and Rational Inequalities
52.
2
1
20
1
x
x
x
x

Interval Test Value Test Conclusion
,1
0
02 false
01
,1
does not belong to the solution set.
53.
43
21
x
x
Express the inequality so that one side is zero.
430
21
x
x

Find the values of x that make the numerator and denominator zero.
570 210
xx
 
Chapter 11 Quadratic Equations and Functions
The boundary points are
17
and
25
.
Interval Test Value Test Conclusion
1
,2




0
04 3
20 1
43, true

1
,2

belongs to the solution set.
54.
11
3
110
3
x
x

Interval Test Value Test Conclusion
,3
0
11 true
,3
belongs to the solution set.
Section 11.5 Polynomial and Rational Inequalities
55.
22
2
x
x
Express the inequality so that one side is zero.
Find the values of x that make the numerator and denominator zero.
60 20
62
6
xx
xx
x
 
 

The boundary points are
6 and 2
.
Interval Test Value Test Conclusion
,6 
−7
72 2
72
92, true


,6 
belongs to the solution set.
56.
2
2
20
2
x
x
x
x

Chapter 11 Quadratic Equations and Functions
40 20
42
4
xx
xx
x
   
 

Interval Test Value Test Conclusion
57.
2
2
252
2520
fx gx
xx
xx


Solve the related quadratic equation.
The boundary points are
1
2
and 2.
2
Interval Test Value Test Conclusion
0
11
20 50 2
,, belongs to the solution set.
22
02, True
 

 
 
 

Section 11.5 Polynomial and Rational Inequalities
58.
2
4; 9 2
fx x gx x

2
492
fx gx
xx

2
Interval Test Value Test Conclusion
11
, 0 4 0 9 0 2 false , is not part of the solution set
44
 
   
 
 
4

59.
21
1
fx gx
x
x
Express the inequality so that one side is zero.
210
1
x
x

Chapter 11 Quadratic Equations and Functions
Interval Test Value Test Conclusion
,1 
3
23 1
31
31, false

,1 
does not belong to the solution set.
60.
;1
21
x
fx gx
x

1
21
fx gx
x
x
Interval Test Value Test Conclusion
101
, 0 1 false , is not part of the solution set
22012
 
  
 
 
2