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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