Analyzing Graphs of Absolute Value Functions

When solving an equation graphically, the first step is to rearrange the equation so that the absolute value term is isolated. $$\begin{array}{c}|x-3|+2=5\iff |x-3|=3\end{array}$$ The left-hand side of the equation can now be expressed as the absolute value function $f(x)=|x-3|.$ We'll draw its graph.

The solutions to the equation are the $x\text{-}$coordinates of the points on the graph that have $y\text{-}$coordinate $3.$

The solutions of the equation are $x=0$ and $x=6.$ We can verify them by substituting in the given equation. We'll start with $x=0.$ Since $x=0$ makes a true statement, it is a solution to the equation. Next, we'll verify $x=6$ in the same way. $$\begin{array}{c}|6-3|+2\stackrel{?}{=}5\Rightarrow 5=5✓\end{array}$$ Since $x=6$ makes a true statement, it is also a solution. Therefore, both $x=0$ and $x=6$ are solutions to the given equation.