Thursday, July 15, 2010

Absolute Value Inequality: A Graphical Approach

An absolute value inequality, such as,

\left| {x + 2} \right| < 3

can be solved as follows:

\begin{array}{c}
  - 3 < x + 2 < 3 \\
  - 3 < x + 2 < 3 \\ 
  - 5 < x < 1 \\ 
 \end{array}

We can visualize these solutions if we graph the function,

f(x) = \left| {x + 2} \right|

in the rectangular coordinate plane and determine where the graph lies below the horizontal line given by y = 3.  In other words, for what x-values is

f(x) < 3

This idea is illustrated below:

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

\left| {x + 2} \right| > 3


We can see that the solutions are,

x <  - 5\,\,\,or\,\,\,x > 1

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.