This section assumes that you have access to a graphing calculator or some other graphing program.

Let's suppose you want to solve the inequality

Here is the graph of the function

A given *x* will solve the inequality if *f*(*x*)<0, *i.e.,* if *f*(*x*) is below the *x*-axis. Thus the set of our solutions is the part of the *x*-axis indicated below in red, the interval (-1,1):

If we want to see the solutions of the inequality

that's just as easy. Now we have to pick all values of

Note the pivotal role played by the "yellow dots", the *x*-intercepts of *f*(*x*).

*f*(*x*) can only change its sign by passing through an *x*-intercept, *i.e.,* a solution of *f*(*x*)=0 will always separate parts of the graph of *f*(*x*) above the *x*-axis from parts below the *x*-axis. This property of polynomials is called the **Intermediate Value Property** of polynomials; your teacher might also refer to this property as **continuity**.

Let us consider another example: Solve the inequality

Here is the graph of the function

A given *x* will solve the inequality if
,
*i.e.,* if *f*(*x*) is above the *x*-axis. Thus the set of our solutions is the part of the *x*-axis indicated below in blue, the union of the following three intervals:

The (finite) endpoints are included since at these points *f*(*x*)=0 and so these *x*'s are included in our quest of finding the solutions of
.

Our answer is approximate, the endpoints of the intervals were found by inspection; you can usually obtain better estimates for the endpoints by using a numerical solver to find the solutions of *f*(*x*)=0. In fact, as you will learn in the next section, the precise endpoints of the intervals are
,
-1, 0 and
.

Two more caveats: The method will only work, if your graphing window contains all *x*-intercepts. Here is a rather simple-minded example to illustrate the point: Suppose you want to solve the inequality

If your graphing window is set to the interval [-5,5], you will miss half of the action, and probably come up with the incorrect answer:

To find the correct answer, the interval (0,10), your graphing window has to include the second *x*-intercept at *x*=10:

Here is another danger: Consider the three inequalities , and . If you do not zoom in rather drastically, all three graphs look about the same:

Only zooming in reveals that the solutions to the three inequalities show a rather different behavior. The first inequality has a single solution, *x*=0. (This also illustrates the fact that a function *f*(*x*) does not always change sign at points where *f*(*x*)=0.)

The second inequality, , has as its solutions the interval [-0.01,0.01]:

The third inequality, , has no solutions:

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