An absolute value represents the distance from a point, in either direction. Because of this, we have to consider the distance in the positive direction and in the negative direction.
|a|=a AND |- a|=a
Therefore, to solve an absolute value equation, we need to think about two cases. The first case being when the expression inside the absolute value symbols is the positive result and the second being when it is the negative result. Let's look at the example |x+5|=8. We have the following two cases.
Case I:& x+5=8
Case II:& x+5=-8
We can find these solutions algebraically, graphically, or numerically.
Algebraically
To solve algebraically, we need to solve both cases like any other equation.
Both x=3 and x=-13 satisfy our absolute value equation.
Graphically
Similarly, for the more basic equations, we can solve this on a number line. We begin by plotting the center point. This is found when the absolute value is in the format
|x-a|
where a is the value plotted on the number line. Ours is at -5.
|x+5|=|x-(-5)|
Then we can move 8 in either direction to find our answers.
Once again, we found the same solutions.
x=3 and x=-13
Numerically
If we want to solve our absolute value numerically, we are just taking a more systematic approach to guessing and checking. We can use a table of values, substitute them into the equation, and see which ones satisfy the equality.
