Tuesday, 26 March 2013

Solving Polynomial Inequalities by Graphing

Let's suppose you want to solve the inequality


Here is the graph of the function f(x)=x2-1:

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


that's just as easy. Now we have to pick all values of x for which f(x)=x2-1 is above the x-axis. As you can see, we obtain as solutions the set $(-\infty,1)\cup(1,\infty)$, 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 theIntermediate Value Property of polynomials; your teacher might also refer to this property as continuity.

Let us consider another example: Solve the inequality

\begin{displaymath}x^4+x^3-2x^2-2x\geq 0.\end{displaymath}

Here is the graph of the function f(x)=x4+x3-2x2-2x>0:

A given x will solve the inequality if $f(x)\geq 0$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:

\begin{displaymath}(-\infty,-1.4]\cup [-1,0] \cup [1.4,\infty)\end{displaymath}

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 $f(x)\geq 0$.

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 $-\sqrt{2}\approx -1.414$, -1, 0 and $\sqrt{2}\approx 1.414$.

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 $x^2 \leq 0$$x^2-0.0001\leq 0$ and $x^2+0.0001\leq 0$. 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, $x^2-0.0001\leq 0$, has as its solutions the interval [-0.01,0.01]:

The third inequality, $x^2+0.0001\leq 0$, has no solutions:

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