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

Recent Articles

  1. Additive Inverse of a Complex Number

    Sep 24, 21 03:39 AM

    What is the additive inverse of a complex number? Definition and examples

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.