# Solving Absolute Value Equations 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.

= 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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