6-7 Solving Radical Equations and Inequalities
Solve each equation.
4.
1.
SOLUTION:
SOLUTION:
5.
2.
SOLUTION:
SOLUTION:
6.
SOLUTION: 3.
SOLUTION:
7. 4.
SOLUTION:
SOLUTION:
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Page 1
6-7 Solving Radical Equations and Inequalities
7.
10.
SOLUTION:
SOLUTION:
8.
Check:
SOLUTION:
Therefore, the equation has no solution.
9.
11.
SOLUTION:
SOLUTION:
10.
SOLUTION: 12.
SOLUTION: eSolutions Manual - Powered by Cognero
Page 2
6-7 Solving Radical Equations and Inequalities
13. CCSS REASONING The time T in seconds that it takes a pendulum to make a complete swing back
12.
and forth is given by the formula
SOLUTION:
, where
L is the length of the pendulum in feet and g is the acceleration due to gravity, 32 feet per second squared.
13. CCSS REASONING The time T in seconds that it takes a pendulum to make a complete swing back and forth is given by the formula
, where
L is the length of the pendulum in feet and g is the acceleration due to gravity, 32 feet per second squared.
a. In Tokyo, Japan, a huge pendulum in the Shinjuku building measures 73 feet 9.75 inches. How long does it take for the pendulum to make a complete swing?
a. In Tokyo, Japan, a huge pendulum in the Shinjuku building measures 73 feet 9.75 inches. How long does it take for the pendulum to make a complete swing?
b. A clockmaker wants to build a pendulum that takes 20 seconds to swing back and forth. How long should the pendulum be?
SOLUTION: a. Convert 73 feet 9.75 inches to feet. 73 feet 9.75 inches =
ft.
Substitute 73.8125 and 32 for L and g then simplify.
b. A clockmaker wants to build a pendulum that takes 20 seconds to swing back and forth. How long should the pendulum be?
Therefore, the pendulum takes about 9.5 seconds to complete a swing.
SOLUTION: a. Convert 73 feet 9.75 inches to feet. 73 feet 9.75 inches =
b. Substitute 20 for T and 32 for g.
ft.
Substitute 73.8125 and 32 for L and g then simplify.
Therefore, the pendulum takes about 9.5 seconds to complete a swing.
b. Substitute 20 for T and 32 for g.
The pendulum should be about 324 ft long.
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14. MULTIPLE CHOICE Solve
Page 3
.
The pendulum should be about 324Inequalities ft long. 6-7 Solving Radical Equations and
Option B is the correct answer.
14. MULTIPLE CHOICE Solve
Solve each inequality.
.
15.
Ay =1
C y = 11
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
B y =5
D y = 15
SOLUTION:
Solve
.
Option B is the correct answer. Solve each inequality.
The solution region is
15.
.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
16.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
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Page 4
The solution region is . 6-7 Solving Radical Equations and Inequalities
The solution region is
.
17.
16.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
.
Solve
The solution region is
.
17.
The solution region is
.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
18.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
The solution region is
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18.
.
The solution region is
.
Page 5
The solution region is . Inequalities 6-7 Solving Radical Equations and
The solution region is
.
18.
19.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
The solution region is
.
The solution region is x > 1.
19.
20.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
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The solution region is x > 1.
The solution region is
.
.
Page 6
The solution region is x > 1. and Inequalities 6-7 Solving Radical Equations
The solution region is
.
20.
21.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
.
Solve
The solution region is
.
The solution region is
.
21.
22.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
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The solution region is
.
Page 7
The solution region is
.
The solution region is
The solution region is . Inequalities 6-7 Solving Radical Equations and
.
Solve each equation. Confirm by using a graphing calculator.
22.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
23.
SOLUTION:
Solve
.
CHECK:
The solution region is
[-2, 28] scl: 2 by [-2, 8] scl: 1
.
Solve each equation. Confirm by using a graphing calculator.
24.
SOLUTION:
23.
SOLUTION:
CHECK:
CHECK:
[-2, 18] scl: 2 by [-2, 18] scl:2
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[-2, 28] scl: 2 by [-2, 8] scl: 1
25.
Page 8
[-2, 28] scl: 2 by [-2, 8] scl: 1
[-2, 18] scl: 2 by [-2, 18] scl:2
6-7 Solving Radical Equations and Inequalities 24.
25.
SOLUTION:
SOLUTION:
CHECK:
CHECK:
[-2, 18] scl: 2 by [-2, 18] scl:2
25.
[-10, 10] scl:1 by [-10, 10] scl:1
SOLUTION:
26.
SOLUTION:
CHECK:
CHECK: eSolutions Manual - Powered by Cognero
[-10, 10] scl:1 by [-10, 10] scl:1
Page 9
[-10, 10] scl:1 by [-10, 10] scl:1 and Inequalities 6-7 Solving Radical Equations
[-2, 18] scl: 2 by [-2, 8] scl: 1
26.
27.
SOLUTION:
SOLUTION:
Check:
CHECK:
[-2, 18] scl: 2 by [-2, 8] scl: 1
[-2, 18] scl: 2 by [-10, 10] scl:1
27.
There is no real solution for the equation.
SOLUTION:
28.
SOLUTION:
Check:
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Check:
Page 10
[-2, 18] scl: 2 by [-10, 10] scl:1
[-2, 18] scl: 2 by [-10, 10] scl:1
There is Radical no real solution for the equation. 6-7 Solving Equations and Inequalities
There is no real solution for the equation.
28.
29.
SOLUTION:
SOLUTION:
CHECK:
Check:
[-1, 4] scl: 1 by [-1, 14] scl: 1
30.
SOLUTION:
[-2, 18] scl: 2 by [-10, 10] scl:1
There is no real solution for the equation.
29.
CHECK:
SOLUTION:
[-5, 45] scl: 5 by [-5, 45] scl: 5
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CHECK:
Page 11
31.
[-1, 4] scl: 1 by [-1, 4] scl: 1
6-7 Solving Radical [-5, 45] scl: 5 by [-5,Equations 45] scl: 5 and Inequalities
33.
31.
SOLUTION:
SOLUTION:
CHECK:
CHECK:
[-2, 18] scl: 2 by [-2, 18] scl:2
32.
SOLUTION: [-1, 9] scl: 1 by [-5, 5] scl: 1
34.
SOLUTION:
CHECK:
[-1, 4] scl: 1 by [-1, 4] scl: 1
33.
SOLUTION: eSolutions Manual - Powered by Cognero
CHECK:
Page 12
[-1, 9] scl:Radical 1 by [-5, 5] scl: 1 6-7 Solving Equations and Inequalities
[-1, 9] scl: 1 by [-5, 5] scl: 1
35. CCSS SENSE-MAKING Isabel accidentally dropped her keys from the top of a Ferris wheel. The
34.
formula
SOLUTION:
describes the time t in seconds
at which the keys are h meters above the ground and Isabel is d meters above the ground. If Isabel was 65 meters high when she dropped the keys, how many meters above the ground will the keys be after 2 seconds?
SOLUTION: Substitute 2 for t and 65 d.
CHECK:
The keys will be 1 meter above the ground after 2 seconds.
Solve each equation.
[-1, 9] scl: 1 by [-5, 5] scl: 1
36.
35. CCSS SENSE-MAKING Isabel accidentally dropped her keys from the top of a Ferris wheel. The formula
SOLUTION:
describes the time t in seconds
at which the keys are h meters above the ground and Isabel is d meters above the ground. If Isabel was 65 meters high when she dropped the keys, how many meters above the ground will the keys be after 2 seconds?
SOLUTION: Substitute 2 for t and 65 d.
37.
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Page 13
SOLUTION:
6-7 Solving Radical Equations and Inequalities
37.
39.
SOLUTION:
SOLUTION:
40.
38.
SOLUTION:
SOLUTION:
41.
39.
SOLUTION:
SOLUTION:
eSolutions Manual - Powered by Cognero
42.
Page 14
40.
6-7 Solving Radical Equations and Inequalities
45.
42.
SOLUTION:
SOLUTION:
46.
43.
SOLUTION:
SOLUTION:
47.
44.
SOLUTION:
SOLUTION:
48. MULTIPLE CHOICE Solve
.
45.
eSolutions Manual - Powered by Cognero SOLUTION:
A 23
B 53
C 123
Page 15
Option D is the correct answer.
6-7 Solving Radical Equations and Inequalities
48. MULTIPLE CHOICE Solve
.
49. MULTIPLE CHOICE Solve
.
A 23
F 41
B 53
G 28
C 123
H 13
D 623
J1
SOLUTION:
SOLUTION:
Option D is the correct answer.
49. MULTIPLE CHOICE Solve
Option F is the correct answer.
.
F 41
Solve each inequality.
G 28
H 13
J1
SOLUTION:
50.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Option F is the correct answer.
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Solve each inequality.
Page 16
The solution region is
Option FRadical is the correct answer. 6-7 Solving Equations and Inequalities
Solve each inequality.
.
51.
50.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
The solution region is
.
52.
The solution region is
.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
51.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
.
Solve
.
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Page 17
The solution region is
.
Since the value of radical is nonnegative, the inequality has no real solution.
The solution region is .and Inequalities 6-7 Solving Radical Equations
52.
54.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve and .
Solve
.
.
Solve
The solution region is
The solution region is
.
.
55.
53.
SOLUTION:
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve and .
Since the value of radical is nonnegative, the inequality has no real solution.
Solve
.
54.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve and .
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Page 18
The solution region is
The solution region is . and Inequalities 6-7 Solving Radical Equations
.
55.
56.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve and .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
.
Solve
Solve
.
The solution region is
.
The solution region is
.
57.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
56.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
Solve
.
.
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The solution region is
Page 19
.
The solution region is
The solution region is . 6-7 Solving Radical Equations and Inequalities
.
57.
58.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
Solve
.
.
The solution region is
.
The solution region is
.
59.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
58. SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
Solve
Solve
.
.
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The solution region is
The solution region is
.
.
Page 20
The solution region is . Inequalities 6-7 Solving Radical Equations and
The solution region is
.
59.
60.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve .
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve . and
.
Solve
Solve
.
The solution region is
.
60.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve . and
The solution region is
.
61.
Solve
.
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve . and
Solve
.
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Page 21
The solution region is . Inequalities 6-7 Solving Radical Equations and
.
61.
62. PENDULUMS The formula
SOLUTION: Since the radicand of a square root must be greater than or equal to zero, first solve . and
represents
the swing of a pendulum, where s is the time in seconds to swing back and forth, and is the length of the pendulum in feet. Find the length of a pendulum that makes one swing in 1.5 seconds.
SOLUTION: Substitute 1.5 for s and solve for l.
Solve
The solution region is
.
The solution region is
.
The length of the pendulum is about 1.82 ft.
62. PENDULUMS The formula
represents
the swing of a pendulum, where s is the time in seconds to swing back and forth, and is the length of the pendulum in feet. Find the length of a pendulum that makes one swing in 1.5 seconds.
63. FISH The relationship between the length and mass of certain fish can be approximated by the equation , where L is the length in meters and M is the mass in kilograms. Solve this equation for M .
SOLUTION:
SOLUTION: Substitute 1.5 for s and solve for l.
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64. HANG TIME Refer to the information at the beginning of the lesson regarding hang time. Describe how the height of a jump is related to the amount of time in the air. Write a step-by-step explanation of how to determine the height of Page 22 Jordan’s 0.98-second jump.
6-7 Solving Radical Equations and Inequalities
64. HANG TIME Refer to the information at the beginning of the lesson regarding hang time. Describe how the height of a jump is related to the amount of time in the air. Write a step-by-step explanation of how to determine the height of Jordan’s 0.98-second jump.
SOLUTION: If the height of a person’s jump and the amount of time he or she is in the air are related by an equation involving radicals, then the hang time associated with a given height can be found by solving a radical equation.
The radius of the region is about 282 ft.
66. WEIGHTLIFTING The formula can be used to estimate the maximum total mass that a weightlifter of mass B kilograms can lift using the snatch and the clean and jerk. According to the formula, how much does a person weigh who can lift at most 470 kilograms?
SOLUTION: Substitute 470 for M and solve for B.
65. CONCERTS The organizers of a concert are preparing for the arrival of 50,000 people in the open field where the concert will take place. Each person is allotted 5 square feet of space, so the organizers rope off a circular area of 250,000 square feet. Using the formula , where A represents the area of the circular region and r represents the radius of the region, find the radius of this region.
SOLUTION: Substitute 250,000 for A and solve for r.
The person weigh 163 kg can lift at most 470 kilograms.
67. CCSS ARGUMENTS Which equation does not have a solution?
The radius of the region is about 282 ft.
66. WEIGHTLIFTING The formula can be used to estimate the maximum total mass that a weightlifter of mass B kilograms can lift using the snatch and the clean and jerk. According to the formula, how much does a person weigh who can lift at most 470 kilograms?
SOLUTION:
SOLUTION: Substitute 470 for M and solve for B.
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Since the value of the radical is negative, it doesPage not 23 have real solution.
Yes; since , the left side of the equation is nonnegative. Therefore, the left side of the equation cannot equal –4. Thus the equation has no solution.
The person weigh 163 kg can lift at most 470 kilograms. 6-7 Solving Radical Equations and Inequalities
67. CCSS ARGUMENTS Which equation does not have a solution?
69. REASONING Determine whether
is
sometimes, always, or never true when x is a real number. Explain your reasoning.
SOLUTION:
SOLUTION:
Since the value of the radical is negative, it does not have real solution.
But this is only true when x = 0. And in that case, we have division by zero in the original equation. So the equation is never true.
68. CHALLENGE Lola is working to solve . She said that she could tell there was no real solution without even working the problem. Is Lola correct? Explain your reasoning.
SOLUTION: Yes; since , the left side of the equation is nonnegative. Therefore, the left side of the equation cannot equal –4. Thus the equation has no solution.
70. OPEN ENDED Select a whole number. Now work backward to write two radical equations that have that whole number as solutions. Write one square root equation and one cube root equation. You may need to experiment until you find a whole number you can easily use.
SOLUTION: Sample answer using
69. REASONING Determine whether
is
sometimes, always, or never true when x is a real number. Explain your reasoning.
SOLUTION:
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Page 24
But this is only true when x = 0. And in that case, we have division by zero in the original equation. So the equation never true.and Inequalities 6-7 Solving RadicalisEquations
They are the same number. For example, .
70. OPEN ENDED Select a whole number. Now work backward to write two radical equations that have that whole number as solutions. Write one square root equation and one cube root equation. You may need to experiment until you find a whole number you can easily use.
72. OPEN ENDED Write an equation that can be solved by raising each side of the equation to the given power.
a.
power
SOLUTION: Sample answer using
b.
power
c.
power
SOLUTION: a. Sample answer:
b. Sample answer:
c. Sample answer:
73. CHALLENGE Solve y = b if and only if x = y.)
for x. (Hint: b
x
SOLUTION:
71. WRITING IN MATH Explain the relationship between the index of the root of a variable in an equation and the power to which you raise each side of the equation to solve the equation.
Equate the powers and solve for x.
SOLUTION: They are the same number. For example, .
REASONING Determine whether the following statements are sometimes, always, or never true for
. Explain your reasoning.
72. OPEN ENDED Write an equation that can be solved by raising each side of the equation to the given power.
eSolutions Manual - Powered by Cognero a.
power
74. If n is odd, there will be extraneous solutions.
SOLUTION: never; Page 25 Sample answer: The radicand can be negative.
Equate the powers and solve for x.
6-7 Solving Radical Equations and Inequalities
REASONING Determine whether the following statements are sometimes, always, or never true for
SOLUTION: sometimes; Sample answer: when the radicand is negative, then there will be extraneous roots.
. Explain your reasoning.
74. If n is odd, there will be extraneous solutions.
76. What is an equivalent form of
A
SOLUTION: never; Sample answer: The radicand can be negative.
B
C
75. If n is even, there will be extraneous solutions.
?
D
SOLUTION: sometimes; Sample answer: when the radicand is negative, then there will be extraneous roots.
SOLUTION:
76. What is an equivalent form of
?
A
B
C
D
SOLUTION:
Option A is the correct answer.
77. Which set of points describes a function?
F {(3, 0), (–2, 5), (2, –1), (2, 9)}
G {(–3, 5), (–2, 3), (–1, 5), (0, 7)}
H {(2, 5), (2, 4), (2, 3), (2, 2)}
J {(3, 1), (–3, 2), (3, 3), (–3, 4)}
SOLUTION: In option F and H, the element 2 has more than one image.
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In option J, the elements 3 and –3 have more than one image. In option G, every element has a unique image. So, it describes a function. Page 26
Option A is the correct answer.
Therefore, option G is the correct answer.
Since the given triangle is an isosceles triangle, the measure of another leg should be 20 inches. Therefore, the measure of the base of the triangle is (56 – 20 − 20) or 16 in.
Option A is the correct answer. 6-7 Solving Radical Equations and Inequalities
77. Which set of points describes a function?
79. SAT/ACT If
, what is the value of x?
F {(3, 0), (–2, 5), (2, –1), (2, 9)}
A4
G {(–3, 5), (–2, 3), (–1, 5), (0, 7)}
B 10
H {(2, 5), (2, 4), (2, 3), (2, 2)}
C 11
J {(3, 1), (–3, 2), (3, 3), (–3, 4)}
D 12
E 20
SOLUTION: In option F and H, the element 2 has more than one image.
SOLUTION:
In option J, the elements 3 and –3 have more than one image. In option G, every element has a unique image. So, it describes a function.
Therefore, option G is the correct answer.
78. SHORT RESPONSE The perimeter of an isosceles triangle is 56 inches. If one leg is 20 inches long, what is the measure of the base of the triangle?
Option A is the correct answer.
Evaluate.
SOLUTION: Since the given triangle is an isosceles triangle, the measure of another leg should be 20 inches. Therefore, the measure of the base of the triangle is (56 – 20 − 20) or 16 in.
80.
SOLUTION:
79. SAT/ACT If
, what is the value of x?
A4
B 10
C 11
D 12
E 20
SOLUTION:
81.
SOLUTION:
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Page 27
6-7 Solving Radical Equations and Inequalities
83. GEOMETRY The measures of the legs of a right
81.
2 2
triangle can be represented by the expressions 4x y 2 2 and 8x y . Use the Pythagorean Theorem to find a simplified expression for the measure of the hypotenuse.
SOLUTION:
SOLUTION:
Substitute
82.
and
for a and b and simplify.
SOLUTION:
Therefore, the measure of the hypotenuse is .
Find the inverse of each function.
84.
83. GEOMETRY The measures of the legs of a right
SOLUTION:
2 2
triangle can be represented by the expressions 4x y 2 2 and 8x y . Use the Pythagorean Theorem to find a simplified expression for the measure of the hypotenuse.
Interchange x and y , then solve for y
SOLUTION:
Substitute
and
for a and b and simplify.
85.
SOLUTION:
Therefore, the measure of the hypotenuse is .
Interchange x and y, then solve for y
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Page 28
6-7 Solving Radical Equations and Inequalities
For each graph,
85.
a. describe the end behavior, SOLUTION:
b. determine whether it represents an odddegree or an even-degree polynomial function, and
Interchange x and y, then solve for y
c. state the number of real zeros.
86.
88. SOLUTION:
SOLUTION: a. The function tends to +∞ as x tends to –∞. The function tends to –∞ as x tends to +∞.
Interchange x and y, then solve for y
b. Since the end behaviors are in opposite directions, the graph represents an odd-degree polynomial. c. Since the graph intersects the x-axis at three points, the number of real zeros is 3.
87.
SOLUTION:
Interchange x and y and solve for y.
89.
SOLUTION: a. The function tends to +∞ as x tends to –∞ and +∞.
b. Since the end behaviors are in the same direction, the graph represents an even-degree polynomial.
For each graph,
a. describe the end behavior,
c. Since the graph does not intersect the x-axis, the Page 29 number of real zeros is 0.
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b. determine whether it represents an odd-
the graph represents an odd-degree polynomial.
c. Since the graph intersects the x-axis at three points, the number of real zeros is 3. 6-7 Solving Radical Equations and Inequalities
c. Since the graph intersects the x-axis at one point, the number of real zeros is 1.
Solve each equation. Write in simplest form.
91.
SOLUTION: 89.
SOLUTION: a. The function tends to +∞ as x tends to –∞ and +∞.
b. Since the end behaviors are in the same direction, the graph represents an even-degree polynomial.
92.
c. Since the graph does not intersect the x-axis, the number of real zeros is 0.
SOLUTION:
90.
93.
SOLUTION: a. The function tends to +∞ as x tends to +∞. The function tends to –∞ as x tends to –∞.
SOLUTION:
b. Since the end behaviors are in opposite directions, the graph represents an odd-degree polynomial.
c. Since the graph intersects the x-axis at one point, the number of real zeros is 1.
Solve each equation. Write in simplest form.
94.
91.
SOLUTION:
SOLUTION: eSolutions Manual - Powered by Cognero
Page 30
6-7 Solving Radical Equations and Inequalities
97.
94.
SOLUTION:
SOLUTION:
95.
SOLUTION:
98.
SOLUTION:
96.
SOLUTION:
97.
SOLUTION:
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Page 31
