# Absolute Value Equation

 Need to know how to solve an absolute value equation like this one? |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! 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.

 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 Solving

Let's solve the problem below together!

|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 as to 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.