The **absolute value** of a number is the distance of that number from 0 on a number line. Distance is expressed as a positive value, so the absolute value of a number is always positive. Absolute value is sometimes called magnitude.

Consider the equation \left|x-2\right|=3. We read this as the distance between some value, x, and 2 is three spaces on a number line.

To solve an absolute value equation that results in two solutions, we begin by isolating the absolute value expression. Once the expression is isolated, we write and solve two separate equations without the absolute value bars: one equation equal to the distance in a positive direction, and one equation equal to the distance in a negative direction.\begin{alignedat}{3}x-2&=3&\qquad\qquad x-2&=-3\\x&=5 &\qquad\qquad x&=-1\end{alignedat}

The solutions to absolute value equations can be verified graphically. Once the absolute value expression is isolated, we can create a table of values for the corresponding absolute value function.

In the previous example, the isolated absolute value expression is \left|x-2\right|, so the corresponding function is y=\left|x-2\right|.

x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|

f\left(x\right) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |

Solve each absolute value equation.

a

\left|x\right|=10

Worked Solution

b

\left|2.5x\right|=5

Worked Solution

c

\left|4x-8\right|=-12

Worked Solution

d

2\left|x+1\right|+3=21

Worked Solution

A machine is used to fill each of several bags with 16 ounces of sugar. After the bags are filled, another machine weighs them. The bag can only weigh 0.3 ounces heavier or lighter than the desired weight, otherwise, the bag is rejected. Write the equation for the heaviest and lightest bag the machine will approve.

Worked Solution

Idea summary

An absolute value equation is an equation where at least one expression contains an absolute value. Consider the absolute value equation

\left|ax+b\right| = k

When k\gt0, an absolute value equation has two solutions. When k=0, an absolute value equation has one solution. When k\lt0, an absolute value equation has no solutions.

Viable solutions for an absolute value equation also depend on context. If a solution is mathematically valid but does not make sense in the context then we say it is non-viable.