When absolute values are used in equations or inequalities, they can be more difficult. The important thing to remember is that the absolute value means the equation inside the bars | | may be positive or negative.
TIP: To solve any absolute value inequalities or equations, it is necessary to have the absolute value section on one side while everything else is on the other side before the equation is split into two equations.
So if you're given: 5+ 3|y - 6| = 13, you first need to clean that up to get the absolute value part all by itself. Clean it up as if the absolute value bars were parentheses you wanted to keep:
5 + 2|y - 6| = 13
2 |y - 6| = 8
|y - 6| = 4
At this point it is just a matter of splitting it into equations and solving for the two possible value of y in the equation.
Inequalities
Inequalities are a little trickier to deal with. In these cases the same rules are followed (getting the absolute value on one side and everything else on the other side of the equation, then split into two different equations - one negative and one positive.) Because of the inequality rule of dividing or multiplying negatives, it is important to flip the sign.
If |3x -4| + 6 ≤ 10, what are the values of x?
The first step is to clean up the equation to get the absolute value by itself on one side.
|3x - 4| + 6 ≤ 10
|3x - 4| ≤ 4
The next step is to rewrite the equation minus the absolute value bars then solve for each - positive and negative.
