SOLVING ABSOLUTE VALUE EQUATIONS WORKSHEET

Solve the following absolute value equations :

1) |3x + 5| = 7

2) |7x| = 21

3) |2x + 5| + 6 = 7

4) |x - 3| + 6 = 6

5) 2|3x +4| = 7

6) 3|5x - 6| - 4 = 5

7) |x² - 4x - 5| = 7

8) 0.5|0.5x| - 0.5 = 2.5

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

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

1. Answer :

|3x + 5| = 7

3x + 5 = 7  or  3x + 5 = -7

3x = 2  or  3x = -12

x = 2/3  or  x = -4

2. Answer :

|7x| = 21

7x = 21  or  7x = -21

x = 3  or  x = -3

3. Answer :

|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

4. Answer :

|x - 3| + 6 = 6

Subtract 6 from each side.

|x - 3| = 0

x - 3 = 0

x = 3

5. Answer :

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

6. Answer :

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

7. Answer :

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

Comparing ax² + bx + c = 0 and x² - 4x + 2 = 0, we get

a = 1, b = -4, c = 2

Quadratic formula :

Substitute a = 1, b = -4 and c = 2.

Therefore x = -2 or 6 or 2 - √2 or 2 + √2.

8. Answer :

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

9. Answer :

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

10. Answer :

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

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