Absolute Value Inequality: A Graphical Approach

An absolute value inequality, such as,

can be solved as follows:

We can visualize these solutions if we graph the function,

in the rectangular coordinate plane and determine where the graph lies below the horizontal line given by .  In other words, for what x-values is

This idea is illustrated below:

Furthermore, we could use the same graph to visualize the solutions to,

We can see that the solutions are,

Now take the time to compare this visualization to that given on Wolfram Alpha.  Remember, it is always helpful to understand the geometric interpretation whenever possible.