Absolute Value

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:

You know that, in order to solve for a variable in an absolute value equation, you set the equation inside the absolute value to both the positive and negative versions of the answer.

What do you do when there are additional terms in the equation, outside of the absolute value expression?

Whenever an absolute value equation has additional terms, outside of the absolute value expression, you use algebra to get rid of those extra terms BEFORE you set the equation equal to both the positive and negative forms of the answer.

You have already learned that the absolute value of a variable tells you the value of the variable, but not whether the variable is positive or negative (and, without more information, you have to assume that it could be either).

Any time you solve for a variable whose absolute value is given, your answer will be two possible answers, as in the following equation (to review, go to the Absolute Value (with variables) lesson).

The absolute value of a number is always positive.  It represents the number's distance from 0, and distance is always positive.

However, when you are working with variables and absolute value, you have to remember that while the absolute value of a number is always positive, the original number may have been negative.

Think about this:

$\mid 6 \mid = 6$

So, if you are given $\mid x \mid = 6$, then $x$ might equal $6$.

But, remeber that:

$\mid ^-6 \mid = 6$

The absolute value of a number is the distance that number is from 0.

That means that 3 has an absolute value of 3. As you can see from the number line below, 3 is three units away from 0.