Solving an Absolute Value Inequality

With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where $\left|x-5\right|=4.$ We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve $\left|x-5\right|=4.$

After determining that the absolute value is equal to 4 at $x=1$ and $x=9,$ we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:

To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in Table 1.

Because $1<x<9$ is the only interval in which the output at the test value is less than 4, we can conclude that the solution to $\left|x-5\right|<4$ is $1<x<9,$ or $\left(1,9\right).$

To use a graph, we can sketch the function $f(x)=\left|x-5\right|.$ To help us see where the outputs are 4, the line $g(x)=4$ could also be sketched as in Figure 11.

We can see the following:

• The output values of the absolute value are equal to 4 at $x=1$ and $x=9.$
• The graph of $f$ is below the graph of $g$ on $1<x<9.$ This means the output values of $f(x)$ are less than the output values of $g(x).$
• The absolute value is less than or equal to 4 between these two points, when $1<x<9.$ In interval notation, this would be the interval $\left(1,9\right).$