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. $\begin{gathered} 2|x+1|+6=8 \quad\Leftrightarrow\quad {\color{#0000FF}{|x+1|}}={\color{#009600}{1}} \end{gathered}$ The left-hand side of the equation can now be expressed as the absolute value function $f(x)={\color{#0000FF}{|x+1|}}.$ 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 ${\color{#009600}{1}}.$

The solutions of the equation are $x=\text{-}2$ and $x=0.$ We can verify them by substituting in the given equation. We'll start with $x=\text{-}2.$ Since $x=\text{-}2$ makes a true statement, it is a solution to the equation. Next, we'll verify $x=1$ in the same way. $\begin{gathered} 2|{\color{#0000FF}{0}}+1|+6\stackrel{?}{=}8 \quad\Rightarrow\quad 8=8 \ {\color{#009600}{\huge{\checkmark}}} \end{gathered}$ Since $x=0$ makes a true statement, it is also a solution. Therefore, both $x=\text{-}2$ and $x=0$ are solutions to the given equation.