In this section, I'll consider quadratic inequalities. I'll solve them using the graph of the quadratic function. I'll also look at other inequalities, which I'll solve using sign charts.
A quadratic function is a function of the form . The graph of a quadratic function is a parabola.
If , the parabola opens up; if , the parabola opens down:
You can use the graph of a quadratic function to solve quadratic inequalities.
Example. Solve the quadratic inequality .
The graph of opens upward, because the coefficient of is . Since
the roots are and .
Therefore, the graph looks like this:
The original inequality asks for the values of x for which the parabola is below the x-axis:
The parabola is below the x-axis for .
Example. Solve the quadratic inequality .
The graph of opens downard, because the coefficient of is -1. Since
the roots are and .
Therefore, the graph looks like this:
The original inequality asks for the values of x for which the parabola is below or on the x-axis. The solution is or .
Warning! You can't put the inequalities and together by writing " ". This says that " ", which is absurd. Rule of thumb: The solution set occupied two shaded pieces on the number line, so two inequalities are required to write the answer.
Example. Solve the quadratic inequality .
The graph of opens upward, because the coefficient of is . The quadratic formula shows that the roots are complex numbers; this means that the graph does not intersect the x-axis. It must look like this:
(I've located the vertex of the parabola --- it's at --- for reference, but it doesn't come into this problem.)
The inequality asks for what values of x the parabola is below or on the x-axis. Since the parabola lies entirely above the x-axis, there are no solutions.
You can also solve inequalities using sign charts.
( or are also okay.) If there are terms on both sides, add or subtract terms to move everything to one side.
Example. Solve .
for ; is undefined for . Set up a sign chart with and
Let . I pick points at random in each of the three intervals: -2, 0, 4. (Pick points which make the calculations simple!) I plug the points into f; for example,
The values I get determine the sign (+ or -) of on each interval. I've marked the signs above the sign chart.
The original inequality asks where is negative. From the sign chart, the solution is .
By the way, it's purely coincidental that the +'s and -'s alternate --- they don't have to do that!
Example. Solve .
First, move everything to one side and combine terms over a common denominator:
for and is undefined for . These are the break points on the sign chart:
The inequality asks where is greater than or equal to 0. The solution is or .
Send comments about this page to: Bruce.Ikenaga@millersville.edu.
Copyright 2008 by Bruce Ikenaga