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: **7 m = 49 or 7m = -49 **

**7 m = 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.

**7 m = -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 *

Follow us on our Instagram page @scholastophere and our YouTube channel for more content and educational videos.