Graphing Absolute Value Inequalities

(Video) Graphing Nonlinear Inequalities

by mathman1024

Here's a video by mathman1024 showing you how to graph an absolute value inequality. The problem he looks at is $y\ge2|x+1|-5$. The steps to draw the graph are pretty much the same as linear inequalities.

1. Look at the inequality symbol to see if the graph is dashed or solid.
2. Draw the graph as it were an equality.
3. Pick a point not on the line to test to see where to shade.

So we first note that $\ge$ is being used so the graph will have a solid line. Next we graph the absolute value equality. This equation has three components to look at. First is the $+1$ inside the absolute value. This will shift our graph to the left by one unit. Next is the $-5$ outside. This will shift our graph down five units. Last, there's a $2$ multiplying the absolute value term. This will tell us what slope the lines will have. Applying these all to the basic absolute value graph, we get:

The final step is to pick a point not on the line to test. Since $(0,0)$ isn't on the line, I will choose that. $$\begin{align} \color{red}{(0)}&\ge2|\color{red}{(0)}+1|-5\\ 0&\ge2|1|-5\\ 0&\ge2-5\\ 0&\ge-3 \end{align}$$

So is zero greater than or equal to negative three? Yup! Therefore, since $(0,0)$ is inside the graph, we shade the inside.

Test Yourself: Graph the inequality $2y<2|x-3|+4$.

Click continue to view the answer.