Let's suppose you want to solve the inequality

*x*

^{2}-1<0.

Here is the graph of the function

*f*(

*x*)=

*x*

^{2}-1:

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

*x*

^{2}-1>0,

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

*x*for which

*f*(

*x*)=

*x*

^{2}-1 is above the

*x*-axis. As you can see, we obtain as solutions the set , indicated below in blue.

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

*f*(

*x*)=

*x*

^{4}+

*x*

^{3}-2

*x*

^{2}-2

*x*>0:

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

*x*

^{2}-10

*x*<0.

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: