In this lesson we will discuss how to solve absolute value equations. Remember absolute value represents the distance between a number and zero. Absolute value equations contain two solutions which are negative and positive. Below will be a set of examples with detailed explanations.
Absolute Value Equation Rules:
If |x| = -a, then x = a or x = -a
Example 1:
|7m| = 49
Looking at the problem above rewrite the example as two different equations without the absolute value operation. Then use the absolute value equation rule to solve both equations.
Rewrite as: 7m = 49 or 7m = -49
7m = 49
Take the first equation and divide both sides of the equal sign by 7 to solve for m.
The positive answer (listed below) has been found. We must now find the negative answer.
m = 7
Now find what x is equal to for the second equation.
7m = -49
Take the second equation (listed above) and divide both sides of the equal sign to solve for m.
m = -7
The final answer for this example should be reported as:
m = 7,-7
Example 2:
|y – 10| = 2
First rewrite this example according to the rule without the absolute value sign.
y – 10 = 2 or y – 10 = -2
Now solve both examples for y and report both answers.
y – 10 = 2
We must first isolate y by moving -10 to the other side of the equal sign. This is accomplished by adding 10 to both sides. When a term is moved from one side of the equal sign to the other the sign in front of the term changes.
y – 10 = 2
+10 +10
y = 12
Now let’s take the second equation
y- 10 = -2
Now isolate y
y – 10 = -2
+10 +10
y = 8
The final answer should be reported as:
y = 12, 8
Example 3:
|7m| + 3 = 70
Rewrite the equation based on the law of absolute value equations.
7m + 3 = 70 or 7m + 3 = -70
Take the first equation and solve for m.
7m + 3 = 70
Isolate m by moving 3 to the other side of the equal sign.
7m + 3 = 70
-3 -3
7m = 67
Using the equation above divide both sides of the equal sign by 7 to isolate m.
Now take the second equation and solve for m.
7m + 3 = -70
-3 -3
7m = -73
Now divide both sides of the equal sign by 7 to solve for m.
The final answer should be reported as
Example 4:
4|n+8| = 56
First rewrite the problem according to the absolute value equation laws without the absolute signs.
4(n+8) = 56 or 4(n+8) = -56
Take the first equation and solve for n. The parentheses are used because the 4 needs to be distributed to the n and 8.
4(n+8) = 56
4n + 32 = 56
-32 -32
n = 6
Now take the second equation and solve for n.
4(n+8) = -56
4n + 32 = -56
-32 -32
n = -22
The final answer should be reported as
n = 6, -22
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