How to solve multi-step absolute value equations

This lesson will show you how to solve multi-step absolute value equations with a couple of good examples. 

Example #1

Solve 4|2x - 1| - 8 = 12

4|2x - 1| - 8 = 12

Add 8 to each side of the equation

4|2x - 1| - 8 + 8 = 12 + 8

4|2x - 1| + 0 = 20

4|2x - 1| = 20

Divide each side by 4

(4÷4)|2x - 1| = 20÷4

|2x - 1| = 5

Rewrite |2x - 1| = 5 as two equations

2x - 1 = 5  or  2x - 1 = -5

2x - 1 + 1 = 5 + 1  or  2x - 1 + 1 = -5 + 1

2x + 0 = 6  or  2x + 0 = -4

2x + 0 = 6  or  2x + 0 = -4

2x = 6  or  2x = -4

x = 3  or  x = -2

Check

4|2x - 1| - 8 = 12

4|2(3) - 1| - 8 = 12

4|6 - 1| - 8 = 12

4|5| - 8 = 12

4(5) - 8 = 12

20 - 8 = 12

12 = 12

x = 3 is indeed a solution

4|2x - 1| - 8 = 12

4|2(-2) - 1| - 8 = 12

4|-4 - 1| - 8 = 12

4|-5| - 8 = 12

4(5) - 8 = 12

20 - 8 = 12

12 = 12

x = -2 is indeed a solution

Example #2

Solve 0.5|1 - 3x| + 1 = 11

0.5|1 - 3x| + 1 = 11

Subtract 1 from each side of the equation

0.5|1 - 3x| + 1 - 1 = 11 - 1

0.5|1 - 3x| + 0 = 10

0.5|1 - 3x| = 10

Divide each side of the equation by 0.5

(0.5÷0.5)|1 - 3x| = 10÷0.5

|1 - 3x| = 20

Rewrite |1 - 3x| = 20 as two equations

1 - 3x = 20  or  1 - 3x = -20

Subtract 1 from each side of the equation

1 - 1 - 3x = 20 - 1  or  1 - 1 - 3x = -20 - 1

-3x = 19  or   -3x = -21

x = -19/3 or x  = 7

Check

0.5|1 - 3(-19/3)| + 1 = 11

0.5|1 + 19| + 1 = 11

0.5|20| + 1 = 11

0.5(20) + 1 = 11

10 + 1 = 11

11 = 11

x = -19/3 is indeed a solution

0.5|1 - 3(7)| + 1 = 11

0.5|1 - 21| + 1 = 11

0.5|-20| + 1 = 11

0.5(20) + 1 = 11

10 + 1 = 11

11 = 11

x = 7 is indeed a solution

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