Absolute Value Equation

Need to know how to solve one of those absolute value equations like the one shown at the right? |x - 4| = 6

Don't be intimidated: Remember, the definition of absolute value is the distance a number is from zero. All this really means is the number must be positive, because a distance has to be positive!

In other words, you must make whatever is inside the absolute values a positive number. Quite simply, you must account for when the value is positive, and when it is negative - there are two separate answers!

Case One: The value is positive. Drop the absolute value and solve!

|x - 4| = 6 becomes x - 4 = 6
Solve to find that x = 10.

Case Two: The value is negative. Change all signs then solve!

-(x - 4) = 6 becomes -x + 4 = 6 (distribute the negative!)
Subtract 4 from each side to get -x = 2
Complete the equation to find that x = -2.

The Answers:

You found the two answers to be x = {-2, 10}. Now you must check them to be sure! You've almost solved your first absolute value equation...

Check x = -2:    |-2 - 4| = 6
                         |-6| = 6
                            6 = 6

Check x = 10:    |10 - 4| = 6
                           |6| = 6
                              6 = 6

Tips and Tricks for Easier Solving

|2x + 5| − 3 = 8

Step 1: Isolate the absolute value

New equation - |2x + 5| = 11 (add 3 to each side)

Step 2: Make two equations, One positive & One negative

Positive	Negative

2x + 5 = +11	2x + 5 = −11
2x = 6	2x = −16
x = 3	x = −8

Step 3: Solve each equation

Notice above, the two answers are x = 3 and x =−8.

If you are curious why I changed the sign of the number on the right instead of the quantity on the left, it was simply to make things easier. In effect, it is the same as dividing both sides by −1.

For an additional challenge, try solving absolute value inequalities. Otherwise, take another look at an absolute value equation.

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