56

Section 2.7 – Polynomial and Rational Inequalities

Definition of a Polynomial Inequality

A polynomial inequality is any inequality that can be put into one of the forms

( ) 0f x ( ) 0f x ( ) 0f x ( ) 0f x

Where f is a polynomial function.

2( ) 5 4f x x x (x = 1, 4)

Procedure for Solving Polynomial Inequalities Example

1. Express the inequality in the form ( ) 0?f x 2 12x x

2 12 0x x

2. Solve ( ) 0f x 2 12 0x x

( 3)( 4) 0x x

3, 4x

3. Locate the boundary

4. Choose one test value

5. Write the solution set

3, 4

1 2

2 0 0 , ax bx c if a x x x x

1 2

2 0 0 ax bx c if a x x x

-3 0 4

+ - +

57

Example

Solve 22 5 12 0x x

Solution

22 5 12 0x x (2 3)( 4) 0x x

2 3 0x 4 0x

32

x 4x 32

, 4 ,

Example

Solve: 3 23 3x x x

Solution

3 2 3 03 xx x

2( 3 ( 0) 3)xx x

1) 03)((2

xx

3 0x

21 0x

3x

21x

, 3     1, 1

3x

1x

-4 0 32

+ - +

-3 -1 0

1

- + - +

58

Rational Inequality

Example

Solve: 21

1xx

Solution

1.2 1 : 10 -1 Conx d

xx x

2 1 0

1( 1) ( 1)x x

xx

012 xx 01x

1x ,11 ,

Example

Solve 5 14x

Solution

5 1 04x

Exception:

0 44x x

04

4 5 1 4x

x x

5 4 0x

1 0x

1x 4, 1

Example

Solve 2 1 53 4

xx

Solution

2 1 5 03 4

xx

Exception: 3 4 0 3 4

34x x x

2 1 5 03 4

3 4 3 4xx

x x

2 1 15 20 0x x

13 21 0x

13 21x

2113

x 21 413 3

, ,

1 0 1

+ +

4 0 1

+

21

13 4

3 0

+

59

Position Function

An object that is falling or vertically projected into the air has its height above the ground, s(t), in feet,

given by

2( ) 16 o os t t v t s

v0 is the original velocity (initial velocity) of the object, in feet per second

t is the time that the object is in motion, in second

s0 is the original height (initial height) of the object, in feet

Example

An object is propelled straight up from ground level with an initial velocity of 80 ft per second. Its height

at time t is modeled by

2( ) 16 80s t t t

Where the height s(t), is measured in feet and the time, t, is measured in seconds. In which time interval

will the object be more than 64 feet above the ground?

Solution

216 80 64t t

216 80 64 0t t

2 16 80 64 0t t

2 5 4 0t t

2 5 4 0t t

( 1)( 4) 0t t

1 0t 4 0t

1t 4t

The time interval [1, 4]

60

Exercises Section 2.7 – Polynomial and Rational Inequalities

1. Solve: 2 7 10 0x x

2. Solve: 22 9 18x x

3. Solve: 2 5 4 0x x

4. Solve: 2 2 0x x

5. Solve: 2 4 12 0x x

6. Solve: 3 23 9 27 0x x x

7. Solve: 3 0x x

8. Solve: 3 23 3x x x

9. Solve 3 2 48x x x

10. Solve:

03

xx

11. Solve:

2 22

xx

12. Solve 2 53 2x

x

13. Solve: 3 24 5

x xx x

14. Solve: 4 2 03 1

x xx x

15. Solve: 2 1 13 3 1

x xx x

16. Solve the inequality 6 114

xx

17. A car can be rented from Basic Rental for $260 per week with no extra charge for mileage.

Continental charges $80 per week plus 25 cents for each mile driven to rent the same car. How

many miles must be driven in a week to make the rental cost for Basic Rental a better deal than

Continental's?

18. If a projectile is launched from ground level with an initial velocity of 96 ft. per sec, its height in

feet t seconds after launching is s feet, where

216 96s t t

When will the projectile be greater than 80 ft. above the ground?

19. A projectile is fired straight up from ground level. After t seconds, its height above the ground is s

ft., where

61

216 220s t t

For what time period is the projectile at least 624 ft. above the ground?

74

Solution Section 2.7 – Polynomial and Rational Inequalities

Exercise

Solve: 2 7 10 0x x

Solution

2 7 10 0x x ( 5)( 2) 0x x

2, 5x

Solution: 2 5x and x

,2 (5, )

Exercise

Solve: 22 9 18x x

Solution

22 9 18 0x x (2 3)( 6) 0x x

2 3 0x 6 0x

32

x 6x 32

, 6

Solution: 32

, 6

Exercise

Solve the inequality: 2 5 4 0x x

Solution

2 5 4 0x x

1,4x

Solution: x < 1; x > 4

,1 4,

0 2 5

+ - +

32

0 6

+ - +

75

Exercise

Solve 2 2 0x x

Solution

2 2 0 2,1x x x

Solution: ( , 2) (1, )

Exercise

Solve 2 4 12 0x x

Solution

2 4 12 0x x

2( 4) ( 4) 4(1)(12)

2(1)x

4 16 48 4 32

2 2 x Complex number

No solution

Exercise

Solve: 3 23 9 27 0x x x

Solution

3 23 9 27 0x x x

2( 3) 9( 3) 0x x x

2

2 2

3 0 3 ( 3)( 9) 0

9 0 9 3

x xx x

x x x

Solution: : , 3

Exercise

Solve 3 0x x

Solution

22 2

0 ( 1) 0

1 0 1 1

xx x

x x x

( 1,0) (1, )

∞ 3 3 ∞

- + +

-1 0 1 2

- + - +

+ - +

0

-2 1

76

Exercise

3 23 3x x x

Solution

3 2 3 03 xx x

2( 3) ( 3 0)x xx

2 ( 3)( -1) 0x x

2 2

3 0 3

-1=0 =1 1

x x

x x x

Solution: (∞ , 3] [1, 1]

Exercise

Solve xxx 4823

Solution

04823 xxx

0482 xxx

0x 0482 xx

12

4814211 x

2

1931x

2

1931 0 1

2

1931

− + − +

Since the symbol is which means the positive sign:

The solution: 1 193 1 1932 2

, 0 ,

-3 -1 0 1

- + - +

77

Exercise

Solve:

03

xx

Solution

03

xx

0, 3x x

,0 3,

Exercise

Solve:

2 22

xx

Solution

2 22

xx

Cond. x ≠ 2

2 2 02

xx

2 2 02

xx

2 2( 2) 0x x

2 2 4 0x x

6 0x

6x , 6 2,

Exercise

Solve 2 53 2x

x

Solution

2 5 03 2x

x

3 23 2

2 5 03 2

xx

xx

2 5(3 2 )0

3 2

x x

x

2 15 10 03 2

x xx

9 13 03 2

xx

0 3

+ +

6 2

+

78

9 13 0x

3 2 0x

9 13x

2 3x

139

x

32

x

Solution: 3 13, , 2 9

Exercise

Solve: 3 24 5

x xx x

Solution

Conditions: x + 4 ≠ 0 x ≠ -4 and x -5 ≠ 0 x ≠ 5

3 2 04 5

x xx x

0 3 0 2 3 32 2

0 4 0 5 4 5 4 5

3 2( 4)( 5) 04 5

x xx xx x

( 5)( 3) ( 4)( 2) 0x x x x

2 23 5 15 ( 2 4 8) 0x x x x x x

2 23 5 15 2 4 8 0x x x x x x

14 7 0x

14 7x

714

1

2x

Solution 1

2, 4 ,5

Exercise

Solve: 4 2 03 1

x xx x

Solution

Conditions: x ≠ -3 and x ≠ 1

4 2 03 1

x xx x

0 4 0 2 4 2 00 3 0 1 3

4 2( 3)( 1) 03 1

x xx xx x

( 1)( 4) ( 3)( 2) 0x x x x

32

13

9

0

- + -

-4 0 1/2 5

+ - + -

79

2 25 4 ( 5 6) 0x x x x

2 25 4 5 6 0x x x x

110 2 0

5x x

Solution: 15

3, 1,

Exercise

Solve: 2 1 13 3 1

x xx x

Solution

Conditions: x ≠ -3 and 13

x

2 1 1 03 3 1

x xx x

2 1 1 03 3

3 31

1 3 3 1x x xx xx

xx

3 1 2 1 3 1 0x x x x

2 26 3 2 1 3 3 0x x x x x x

2 26 1 4 3 0x x x x

25 5 4 0x x

.5 1.55 105

10x

-3

5 105

10

13

0

5 105

10

+ - + - +

5 105 5 1051310 10

, 3 , ,

Exercise

Solve the inequality 6 114

xx

Solution

Restriction: 14 0 14x x

6 1 014

xx

0

3 1/5 1

+ +

-

80

14 14 16 1 4014

xx

x xx

6 14 0x x

20 0 Implossible No Solution 6 6 3 101 1 114 14 7

070

0 14

+

Solution: 14,

Exercise

A car can be rented from Basic Rental for $260 per week with no extra charge for mileage.

Continental charges $80 per week plus 25 cents for each mile driven to rent the same car. How many

miles must be driven in a week to make the rental cost for Basic Rental a better deal than Continental's?

Solution

x: number of miles driven

For Continental, cost: 80 + .25x

Basic Rental a better deal than Continental's

260 < 80 + 0.25 x

260 - 80 < 0.25 x

180 < .25 x

720 < x

Solution: more than 720 miles per week.

Exercise

If a projectile is launched from ground level with an initial velocity of 96 ft per sec, its height in feet t

seconds after launching is s feet, where

216 96s t t

When will the projectile be greater than 80 ft above the ground?

Solution

Projectile be greater than 80 ft above the ground

80s

216 96 80t t

216 96 80 0t t

62

116 9

1 60

16 806

t t

2 6 5 0t t

81

2 6 5 0t t

( 1)( 5) 0t t

1, 5t

Solution 1, 5

Exercise

A projectile is fired straight up from ground level. After t seconds, its height above the ground is s ft,

where

216 220s t t

For what time period is the projectile at least 624 ft above the ground?

Solution

Projectile at least 624 ft. 624s

216 220 624t t 216 220 624 0t t Divide by -4

24 55 156 0t t

2( 55) ( 55) 4(4)(156) 55 23

162(4)t

55 23

16t

55 2316

t

7816

3216

398

2

Solution: 398

2,

0 1 5

+ - +