SOLVING ABSOLUTE VALUE EQUATIONS WORKSHEET PDF

Problem 1 :

Solve the absolute value equation :

|3x + 5|  =  7

Problem 2 : 

Solve the absolute value equation :

|7x|  =  21

Problem 3 :

Solve the absolute value equation :

|2x + 5| + 6  =  7

Problem 4 :

Solve the absolute value equation : 

|x - 3| + 6  =  6

Problem 5 :

Solve the absolute value equation :

2|3x +4|  =  7

Problem 6 :

Solve the absolute value equation :

3|5x - 6| - 4  =  5

Problem 7 :

Solve the absolute value equation :

|x² - 4x - 5|  =  7

Problem 8 :

Solve the absolute value equation :

0.5|0.5x| - 0.5  =  2.5

Problem 9 :

If the absolute value equation |2x + k|  =  3 has the solution x  =  -2, find the value of k. 

Problem 10 :

If the absolute value equation |x - 3| - k = 0 has the solution x  =  -5, find the value of k. 

Detailed Answer Key

Problem 1 :

Solve the absolute value equation :

|3x + 5|  =  7

Solution :

Solve :

|3x + 5|  =  7

3x + 5  =  7     or     3x + 5  =  -7 

3x  =  2     or     3x  =  -12

x  =  2/3     or     x  =  -4

So, the solution is x  =  -4, 2/3.

Problem 2 : 

Solve the absolute value equation :

|7x|  =  21

Solution :

Solve :

|7x|  =  21

7x  =  21     or     7x  =  -21 

x  =  3     or     x  =  -3

So, the solution is x  =  -3, 3

Problem 3 :

Solve the absolute value equation :

|2x + 5| + 6  =  7

Solution :

Solve :        

|2x + 5| + 6  =  7

Subtract 6 from each side.

|2x + 5|  =  1

2x + 5  =  1     or     2x + 5  =  1 

2x  =  -4     or     2x  =  -6

x  =  -2     or     x  =  -3

So, the solution is x  =  -2, -3.

Problem 4 :

Solve the absolute value equation : 

|x - 3| + 6  =  6

Solution :

Solve :        

|x - 3| + 6  =  6

Subtract 6 from each side.

|x - 3| + 6  =  6

Subtract 6 from each side.

|x - 3|  =  0

x - 3  =  0     or     x - 3  =  0 

x  =  3     or     x  =  3

So, the solution is x  =  3, 3.

Problem 5 :

Solve the absolute value equation :

2|3x +4|  =  7

Solution :

Solve :              

2|3x +4|  =  7

Divide each side by 2. 

|3x + 4|  =  7/2

3x + 4  =  7/2     or     3x + 4  =  -7/2

3x  =  7/2 - 4     or     3x  =  -7/2 - 4

3x  =  -1/2     or     3x  =  -15/2

x  =  -1/6     or     x  =  -15/6

x  =  -1/6     or     x  =  -5/2

So, the solution is x  =  -1/6, -5/2.

Problem 6 :

Solve the absolute value equation :

3|5x - 6| - 4  =  5

Solution :

Solve : 

3|5x - 6| - 4  =  5

Add 4 to each side.

3|5x - 6|  =  9

Divide each side by 3.

|5x - 6|  =  3

5x - 6  =  3     or     5x - 6  =  -3

5x  =  9     or     5x  =  3

x  =  9/5     or     x  =  3/5

So, the solution is x  =  9/5, 3/5.

Problem 7 :

Solve the absolute value equation :

|x² - 4x - 5|  =  7

Solution :

Solve :

|x² - 4x - 5|  =  7

x² - 4x  - 5  =  7     or     x² - 4x - 5  =  -7 

x² - 4x  - 12  =  0     or     x² - 4x + 2  =  0 

Solve the first quadratic equation x² - 4x - 12  =  0.

x² - 4x  - 12  =  0

(x + 2)(x - 6)  =  0

x + 2  =  0     or     x - 6  =  0

x  =  -2     or     x  =  6 

Solve the second quadratic equation x² - 4x + 2  =  0.

This quadratic equation can not be solved using factoring. Because the left side part can not be factored.

So, we can use quadratic formula and solve the equation as shown below. 

So, the solution is x  =  -2, 7, 2 ± √2.

Problem 8 :

Solve the absolute value equation :

0.5|0.5x| - 0.5  =  2.5

Solution :

Solve :

0.5|0.5x| - 0.5  =  2.5

Add 0.5 to each side. 

0.5|0.5x|  =  3

Divide each side by 0.5

|0.5x|  =  6

0.5x  =  6     or     0.5x  =  -6

x  =  12     or     x  =  -12

So, the solution is x  =  -12, 12.

Problem 9 :

If the absolute value equation |2x + k|  =  3 has the solution x  =  -2, find the value of k. 

Solution :

Because x  =  -2 is a solution, substitute x  =  -2 in the given absolute value equation. 

|2(-2) + k|  =  3

|-4 + k|  =  3

Solve for k :

-4 + k  =  3     or     -4 + k  =  -3

k  =  7     or     k  =  1

So, the value of k  =  1, 7.

Problem 10 :

If the absolute value equation |x - 3| - k = 0 has the solution x  =  -5, find the value of k. 

Solution :

Because x  =  -2 is a solution, substitute x  =  -2 in the given absolute value equation. 

|-5 - 3| - k  =  0

|-8| - k  =  0

8 - k  =  0

8  =  k

So, the value of k  =  8.

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