Solved Examples and Worksheet for Solving and Graphing Absolute Value Equations

Q1Simplify 8 + 12m - 10.
A. 12m - 2
B. 2m
C. 2m + 12
D. None of the above

Solutions
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Step: 1
8 + 12m - 10 = 12m + 8 - 10
  [Arrange like terms.]
Step: 2
= 12m + (8 - 10)
  [Use associative property.]
Step: 3
= 12m - 2
  [Simplify like terms.]
Correct Answer is :   12m - 2
Q2Simplify 4a + b + 3a - 5b.

A. 4ab
B. 7a - 4b
C. 7a - 5b
D. 7b + 4a

Solutions
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Step: 1
4a + b + 3a - 5b
  [Original expression.]
Step: 2
= 4a + 3a + b - 5b
  [Arrange like terms.]
Step: 3
= (4a + 3a) + (b - 5b)
  [Use associative property.]
Step: 4
= 7a - 4b
  [Combine like terms.]
Correct Answer is :   7a - 4b
Q3Simplify:
268m - 32m
A. 268m
B. 236m
C. 236
D. 300m

Solutions
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Step: 1
268m - 32m
  [Original expression.]
Step: 2
(268 - 32)m
  [Use distributive property.]
Step: 3
= 236m
  [Subtract.]
Correct Answer is :   236m
Q4Solve the equation.
| x | = 23

A. 23, - 23
B. 0, - 23
C. - 23
D. 23

Solutions
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Step: 1
| x | = 23
Step: 2
The numbers that are 23 units from zero are 23 and - 23.
Step: 3
There are two solutions, which are 23 and - 23.
Correct Answer is :   23, - 23
Q5Solve the equation.
| x | = - 2
A. 2
B. no solution
C. - 2 , 2
D. - 2

Solutions
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Step: 1
| x | = - 2
  [Original equation.]
Step: 2
The absolute value of a number is always a positive number.
Step: 3
So, the equation | x | = - 2 has no solution.
Correct Answer is :   no solution
Q6Simplify:
11 + 20m - 14
A. 20m - 3
B. 3m + 20
C. 20m + 3
D. 3m

Solutions
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Step: 1
11 + 20m - 14 = 20m + 11 - 14
  [Arrange like terms.]
Step: 2
= 20m + (11 - 14)
  [Use associative property.]
Step: 3
= 20m - 3
  [Simplify like terms.]
Correct Answer is :   20m - 3
Q7Choose the values of x, that makes | 4x + 9 | = 8 true.

A. 14, 174
B. - 14, - 174
C. - 4 , 174
D. - 14, 174

Solutions
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Step: 1
As, | 4x + 9 | = 8, then the values of the expression 4x + 9 is equal to 8 or - 8.
Step: 2
4x + 9 = 8 or 4x + 9 = - 8
Step: 3
4x + 9 - 9 = 8 - 9 or 4x + 9 - 9 = - 8 - 9
  [Subtract 9 from each side.]
Step: 4
4x = -1 or 4x = - 17
  [Simplify.]
Step: 5
x = - 14 or x = - 174
  [Divide by 4 on each side.]
Step: 6
The solutions of the equation are - 14 and - 174.
Correct Answer is :   - 14, - 174
Q8Choose the values of x that makes | x - 8 | - 5 = - 0.35 true.
A. 11.65, 2.35
B. 12.65, 2.35
C. 12.65, 3.35
D. 11.65, 3.35

Solutions
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Step: 1
| x - 8 | - 5 = - 0.35
  [Original equation.]
Step: 2
| x - 8 | - 5 + 5 = - 0.35 + 5
  [Add 5 on both sides.]
Step: 3
| x - 8 | = 4.65
  [Simplify.]
Step: 4
| x - 8 | = 4.65, then x - 8 equals 4.65 or - 4.65
Step: 5
x - 8 = 4.65 or x - 8 = - 4.65
Step: 6
x - 8 + 8 = 4.65 + 8 or x - 8 + 8 = - 4.65 + 8
  [Add 8 on both sides.]
Step: 7
x = 12.65 or x = 3.35
  [Simplify.]
Step: 8
The solutions of the equation are 12.65 and 3.35.
Correct Answer is :   12.65, 3.35
Q9Find the values of x, if | 5x | = 2.85.

A. 0.06, - 0.06
B. 5.7, - 5.7
C. 1.57, - 1.57
D. 0.57, - 0.57

Solutions
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Step: 1
| 5x | = 2.85
  [Original equation.]
Step: 2
As, | 5x | = 2.85 then the values of the expressions 5x equals to 2.85 or - 2.85
Step: 3
5x = 2.85 or 5x = - 2.85
Step: 4
x = 0.57 or x = - 0.57
  [Divide by 5 on both sides.]
Step: 5
The solutions of the equation are 0.57 and - 0.57.
Correct Answer is :   0.57, - 0.57
Q10Which number is a solution of | x | - 6 = 7?
A. - 13
B. 7
C. - 7
D. 1

Solutions
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Step: 1
| x | - 6 = 7
  [Original equation.]
Step: 2
| x | = 13
  [Add 6 on both sides and simplify.]
Step: 3
x = 13 or x = - 13
Step: 4
Among the choices, - 13 is the solution of the equation.
Correct Answer is :   - 13
Q11Solve the absolute value equation.
          | x | = - 2

A. - 2, 2
B. 2
C. no solution
D. - 12, 12

Solutions
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Step: 1
| x | = - 2
  [Original equation.]
Step: 2
The absolute value of a positive or negative number is always a positive number.
Step: 3
The equation has no solution.
Correct Answer is :   no solution
Q12Find the values of x, if | x - 2 | = 10.
A. 8, 10
B. 12, - 12
C. 12, - 8
D. - 8, - 9

Solutions
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Step: 1
| x - 2 | = 10
  [Original Equation.]
Step: 2
x - 2 = 10 (or) x - 2 = - 10
Step: 3
x - 2 + 2 = 10 + 2 (or) x - 2 + 2 = - 10 + 2
  [Add 2 to each side.]
Step: 4
x = 12 (or) x = - 8
  [Simplify.]
Step: 5
The solutions of the equation are 12 and - 8.
Correct Answer is :   12, - 8
Q13Which of the following graphs represents the solution for the equation: |x - 3| = 7?

A. Graph 3
B. Graph 4
C. Graph 1
D. Graph 2

Solutions
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Step: 1
|x - 3| = 7
Step: 2
x - 3 = 7 or x - 3 = - 7
  [Write it as disjunction.]
Step: 3
x = 10 or x = - 4
Step: 4
Check the answer: x = 10
Step: 5
|10 - 3| = 7
Step: 6
7 = 7, which is true.
Step: 7
So, x = 10 is the solution.
Step: 8
Check the answer: x = - 4
Step: 9
|- 4 - 3| = 7
Step: 10
|- 7| = 7
Step: 11
7 = 7, which is true.
Step: 12
So, x = - 4 is the solution.
Step: 13
The graph for the solution:
Correct Answer is :   Graph 2
Q14Which of the following graph represent the solution for the equation: |x + 7| = 9?


A. Graph 1
B. Graph 2
C. Graph 3
D. Graph 4

Solutions
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Step: 1
|x + 7| = 9
Step: 2
x + 7 = 9 or x + 7 = - 9
  [Write it as disjunction.]
Step: 3
x = 2 or x = - 16
Step: 4
Check the answer: x = 2
Step: 5
|2 + 7| = 9
Step: 6
9 = 9, which is true.
Step: 7
So, x = 2 is the solution.
Step: 8
Check the answer: x = - 16
Step: 9
|- 16 + 7| = 9
Step: 10
|- 9| = 9
Step: 11
9 = 9, which is true.
Step: 12
So, x = - 16 is the solution.
Step: 13
The graph for the solution is:
Correct Answer is :   Graph 2
Q15Which of the following graphs represents the solution for the equation?
|x| - 2 = 5


A. Graph 3
B. Graph 2
C. Graph 1
D. Graph 4

Solutions
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Step: 1
|x| - 2 = 5
Step: 2
|x| = 7
  [Add 2 to both sides of the equation.]
Step: 3
x = 7 or x = - 7
  [Write it as disjunction.]
Step: 4
Check the answer: x = 7
Step: 5
|7| - 2 = 5
Step: 6
5 = 5, which is true.
Step: 7
Hence, x = 7 is the solution.
Step: 8
Check the answer: x = - 7
Step: 9
|- 7| - 2 = 5
Step: 10
5 = 5, which is true.
Step: 11
So, x = - 7 is also the solution.
Step: 12
The graph for the solution is:
Correct Answer is :   Graph 4

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