How to Simplify Absolute Values

Deal with an argument that is just a variable. If your argument is just a variable by itself, set equal to a number, then simplifying is very easy. Since absolute value represents a distance from 0, your variable could be either the positive number to which it is equal, or it could be the negative version of that number. There is no way to tell, so include both possibilities in your solution.

• For example, say you know that the absolute value of a variable, x, is equal to 3. You cannot tell if x is positive or negative; you are looking for any number whose distance is 3 from 0. Therefore, you solution is either 3 or -3.
• If this is the kind of argument you need to simplify, stop here. Your work is done. If, however, you have an inequality, continue on.

Recognize absolute value inequalities. If, however, you are given an argument with a variable, expressed as an inequality, more steps are required. Interpret these inequalities as asking you to find all the possible numbers that could work.

• For example, say you have the following inequality.
  This can be interpreted as “Show all numbers whose absolute value is less than 7.” In other words, find all numbers whose distance is 7 from 0, excluding 7 itself. Note that the inequality is constructed as “less than” rather than “less than or equal to.” If it were the later, then 7 itself would be included.

Graph a number line. The first thing to do, when faced with an absolute value inequality, is to draw a number line. Tag points corresponding with the numbers with which you are working.

• In the example above, your number line would look like this.
  The open circles indicate numbers excluded from your final result. Remember: if the inequality were stated as “greater than or equal to” or “less than or equal to,” then these numbers would be included instead. In that case, the circles would be solid.

Consider the numbers on the left side of the number line. Since you don't know whether your variable is positive or negative, you are really dealing with two possible ranges of numbers: those on the left side of the number line and those on the right. First, consider the numbers on the left. Make the variable negative, and convert your absolute value bars to parentheses. Solve.

• In the example above, you would convert the absolute value bars to parentheses to show that (-x) is less than 7. Multiply both sides of the inequality by -1. Note that when you multiply by a negative number, you must switch the inequality sign (from less than to greater than, or vice versa). Your inequality would look like this.
  
  You now know that for the left side of the number line, x will be greater than -7. On a number line, that would look like this.

Consider the numbers on the right side of the number line. Now you can look at the other range of numbers, those that are positive. This is even simpler: make the variable positive, convert your absolute value bars to parentheses.

• In the example above, you would convert the absolute value bars to parentheses to show that (x) is less than 7. No further work is necessary for this step. On a number line, that would look like this.