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. $$\begin{array}{c}\left|\frac{5}{2}(x-2)\right|+4=1\iff \left|\frac{5}{2}(x-2)\right|=\text{-}3\end{array}$$ Recall that an absolute value expression cannot be negative. Therefore, the given function has no solutions. Let's also verify our conclusion by graphing. The left-hand side of the equation can be expressed as the absolute value function $f(x)=\left|\frac{5}{2}(x-2)\right|.$ We'll draw its graph.

The solutions to the equation would be the $x\text{-}$coordinates of the points on the graph that whose $y\text{-}$coordinate is $\text{-}3.$

As we can see above, there is no point on the graph whose $y\text{-}$coordinate is $\text{-}3.$ Therefore, the equation has no solutions.