           ## Absolute Value

Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative.

• The absolute value of 5 is 5.
• distance from 0: 5 units • The absolute value of 5 is 5.
• distance from 0: 5 units • The absolute value of 2 + 7 is 5.
• distance of sum from 0: 5 units • The absolute value of 0 is 0. (This is why we don't say that the absolute value of a number is positive: Zero is neither negative nor positive.)

The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value.

• |6| = 6 means the absolute value of 6 is 6.
• | 6| = 6 means the absolute value of 6 is 6.
• | 2 - x| means the absolute value of 2 minus x.
• |x| means the negative of the absolute value of x.

The number line is not just a way to show distance from zero, it's also a good way to graph absolute value.

Consider |x| = 2. To show x on the number line, you need to show every number whose absolute value is 2. Now think about |x| > 2. To show x on the number line, you need to show every number whose absolute value is greater than 2. When you graph on the number line, an open dot indicates that the number is not part of the graph. The > symbol indicates that the number being compared is not included in the graph. In general, you get two sets of values for inequalities with |x| > some number or with |x| = some number.

Now think about |x| = 2. You are looking for numbers whose absolute values are less than or equal to 2. It turns out that all real numbers from 2 through 2 make the inequality true. When you graph on the number line, a closed dot indicates that the number is part of the graph. The = symbol indicates that the number being compared is included in the graph. In general, you get one set of values for inequalities with|x| < some number or with |x| = some number. An easy way to write these kinds of inequalities to show that their values fall between two numbers is:

• For |x| < 2, 2 < x < 2
• For |x| = 4, 4 = x = 4
• For |x + 6| < 25, 25 < x + 6 < 25

Of course, with less than inequalities, |x| will never be less than 0, so even though x can be negative, the number you're comparing it to can't be (or there won't be any points graphed on your number line).