Using a Graphical Approach to Solve Absolute Value Inequalities

We are trying to determine where $f(x)<0,$ which is when $-\frac{1}{2}|4x-5|+3<0.$ We begin by isolating the absolute value.

Next we solve for the equality $\left|4x-5\right|=6.$

Now, we can examine the graph of $f$ to observe where the output is negative. We will observe where the branches are below the x-axis. Notice that it is not even important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $x=-\frac{1}{4}$ and $x=\frac{11}{4}$ and that the graph has been reflected vertically. See Figure 12.

We observe that the graph of the function is below the x-axis left of $x=-\frac{1}{4}$ and right of $x=\frac{11}{4}.$ This means the function values are negative to the left of the first horizontal intercept at $x=-\frac{1}{4},$ and negative to the right of the second intercept at $x=\frac{11}{4}.$ This gives us the solution to the inequality.

In interval notation, this would be $\left(-\infty ,-0.25\right)\cup \left(2.75,\infty \right).$