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Chapter 3: Problem 59

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x-2}{x+2} \leq 2 $$

Short Answer

Expert verified
The solution to the inequality is \((-鈭�, -2) \cup (-6, 鈭�)\)

Step by step solution

01

Simplify the rational inequality

In order to simplify the rational inequality \(\frac{x-2}{x+2} \leq 2\), subtract 2 from both sides to get it in a more recognizable form and to make the right side into 0. Thus, the inequality becomes \(\frac{x-2}{x+2} - 2 \leq 0\). This can be further simplified to \(\frac{x-2 - 2(x+2)}{x+2} \leq 0\) which gives us \(\frac{-x-6}{x+2} \leq 0\).
02

Find the zero points

To find the zero points isolate x on one side of the inequality. This gives x = -6 or x = -2.
03

Test the intervals

The number line is split into intervals by the zeroes. So test each interval by taking a point from each interval and substituting it back into the simplified inequality. Let's pick points -7, -3 and 0 from the intervals (-鈭�, -2), (-2, -6) and (-6, 鈭�) respectively. For -7, the inequality becomes \(\frac{-(-7)-6}{-7+2} \lessapprox 0\). For -3, it becomes \(\frac{-(-3)-6}{-3+2} \lessapprox 0\), and for 0, it becomes \(\frac{-0 - 6}{0+2} \lessapprox 0\). The first and the third inequalities are true
04

Express the solution in interval notation

The solution on the number line are the intervals (-鈭�, -2) and (-6, 鈭�). When expressed in interval notation it is: \((-鈭�, -2) \cup (-6, 鈭�)\) . Note that the parenthesis means the endpoints are not included in the solution set. If a bracket was used then it would mean the endpoint is included.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations $$ \text { of } f(x)=\frac{1}{x} \text { to graph } g $$. $$ g(x)=\frac{2 x+7}{x+3} $$

Although I have not yet learned techniques for finding the \(x\) -intercepts of \(f(x)=x^{3}+2 x^{2}-5 x 6,\) I can easily determine the \(y\) -intercept.

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has vertical asymptotes given by x=-2 and x=2, a horizontal asymptote y=2, y -intercept at \frac{9}{2}, x -intercepts at -3 and 3, and y -axis symmetry.

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomial鈥檚 degree, and turning points.

Can the graph of a polynomial function have no x-intercepts? Explain.
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