Absolute Value

The absolute value of a number a, denoted |a|, is the distance between a and 0 on the number line. Since 4 is 4 units to the right of 0 on the number line and –4 is 4 units to the left of 0 on the number line, both have an absolute value of 4:

Since 4 and –4 are the only numbers that are 4 units from 0, if |x| = 4, then x = 4 or x = –4. If |x| < 4, then x is less than 4 units from 0, which means –4 < x < 4. If |x| > 4, then x is more than 4 units from 0, which means either that x < –4 or x > 4.

Key Fact A2

If b is a positive number, then

These results are displayed graphically on the three number lines below.

|x| = b
Number line with -b, 0, and b. Points at -b and b are filled in dots.
|x| < b
Number line with -b, 0, and b. Line range from -b and b, which have open circles.
|x| > b
Number line with -b, 0, and b. Open circle at point -b has line pointing left, and open circle at point b has line pointing right.

Given any two numbers x and y, you can always find their sum (x + y), difference (x – y), product (xy), and quotient (x ÷ y)—with your calculator, whenever necessary—except that you may never divide by zero. For example, 4 ÷ 0 is meaningless. If you attempt to divide by 0 on your calculator, you will get an error message.