Section 2.7 Polynomial and Rational Inequalitiesmrsk.ca/AP/polynRationalIneq.pdf · 56 Section 2.7...
Transcript of Section 2.7 Polynomial and Rational Inequalitiesmrsk.ca/AP/polynRationalIneq.pdf · 56 Section 2.7...
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
- + - +
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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]
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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
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216 220s t t
For what time period is the projectile at least 624 ft. above the ground?
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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
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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
- + - +
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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
+ - + -
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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
+ - +