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**