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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