Absolute Value Inequalities

Just as there are absolute value equations, there are also absolute value inequalities.  You solve absolute value inequalities in the exact same way that you deal with absolute value equations, but with the twist that is similar to the twist involved with regular inequalities: you have to flip the inequality sign when you flip the signs of the numbers.

To solve an absolute value inequality:

1. Simplify the inequality, getting rid of any extra terms, until you have just the absolute value expression on one side of the inequality sign, and a number or expression on the other side.
2. Write two separate inequalities.
3. One inequality will be exactly the same as the original problem, but without absolute value signs.
4. The second inequality will have the inequality sign reversed, and the answer will have the opposite sign as the original (so, positive if it was negative, or negative if it was positive).
5. Solve as you would any other inequality.

Let's see an example:

$\mid x + 3\mid -7 \geq 5$

Before we can deal with the absolute value, we need to get rid of the $^-7$.

$$\eqalign{\mid x + 3\mid -7&\geq 5\\+7 \; &\; +7\\ \mid x+3 \mid & \geq 12}$$

Now, you have a basic absolute value inequality, with an absolute value expression isolated on one side, and a value on the other.  Now, you can write two different inequalities, to solve for $x$.

$$\eqalign {\mid x + 3 \mid & \geq 12\\x+3& \geq 12 \text{ or } \leq -12}$$

$$\begin {array}{cc}\eqalign{\mathbf{x+3}&\mathbf{\geq 12} \\ -3&\quad -3 \\x&\geq 9 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{\mathbf{x+3}&\mathbf{\leq ^-12}\\ -3&\quad -3\\x& \leq -15}\end{array}$$

$x \geq 9 \text{ or } x \leq -15$

No matter how your additional terms are attached to your absolute value expression, you need to use algebra to get rid of them, and isolate your absolute value expression, before you write your two inequalities and solve for the variable. Remember, when you are solving the inequalities, if you multiply or divide by a negative number, you need to flip the sign again.

The rule with any equation that contains an absolute value: isolate the absolute value expression (just like it was a variable) and then, once you find the value of the absolute value expression, write your two (one positive, and one negative, one with a greater than sign and one with a less than sign) inequalities, and solve.