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Chapter 2: Problem 49

Solve the inequality. Then graph the solution set. $$\frac{x^{2}+2 x}{x^{2}-9} \leq 0$$

Short Answer

Expert verified
The solution to the inequality is x \(\in\) (-2, 3]

Step by step solution

01

Simplify

First, we need to make the expression a bit easier to work with. The inequality \(\frac{{x^{2}+2 x}}{{x^{2}-9}} \leq 0\) can be rewritten as \(\frac{{x(x + 2)}}{{(x-3)(x+3)}} \leq 0\), factoring the expressions in the numerator and denominator.
02

Find Critical Values

The critical values are the values of x for which the expression equals zero or is undefined. This will be when the numerator equals zero, or the denominator equals zero. The critical values are thus x = -2, 3, -3.
03

Test Intervals

Now we'll test the intervals between our critical values. We'll pick a test value from each interval and substitute it into our inequality to see if it's true or false. For our inequality, intervals are x < -3; -2 < x < 3; x > 3. The test points will be x = -4, 0, and 4.
04

Sign Analysis

By substituting the values from step 3, we can determine where the function is positive or negative. The expression is negative when x is in interval (-2,3), and positive elsewhere. Hence, the solutions would be where the expression is negative and zero. As the inequality is less than or equal to zero, the solution is x \(\in\) (-2, 3].
05

Graph the Solution

On a number line, indicate the solution interval as a solid line from -2 to 3, filled circle on 3 as it is included and an open circle at -2 as it's not included.

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Most popular questions from this chapter

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$f(x)=x^{4}+6 x^{2}-27$$

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{3}-x^{2}+4 x-4\) Zero \(2 i\)

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=x^{3}+24 x^{2}+214 x+740$$

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.
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