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Chapter 2: Problem 44

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{x+5}{x-2}>0$$

Short Answer

Expert verified
The solution to the inequality \(\frac{x+5}{x-2}>0\) is \(-5 < x < 2\), which is expressed in interval notation as \((-5, 2)\).

Step by step solution

01

Find the Critical Values

Critical values are the values of x that make the expression equal to zero or are undefined. We set each part of the fraction (numerator and denominator) equal to zero and solve for x. For \(x+5=0\), solving for x gives -5 and for \(x-2 = 0\), solving for x gives 2. However, x can never be equal to 2 because it would make the denominator zero and thus undefined.
02

Test the intervals

Next, we divide up the number line using the critical values into three intervals: \(-\infty, -5\), \(-5, 2\), and \(2, \infty\). Pick a test point from each interval and substitute it into the inequality to see if the inequality holds. For example, for interval \(-\infty, -5\), you might choose -6. If the inequality holds then that interval is part of the solution.
03

Express in interval notation

Interval notation is a way of writing subsets of the real number line. An interval is written as \(a, b\), where a and b are the endpoints. Open parentheses indicate that the endpoint is not included in the interval, and a closed parenthesis indicates it is included. For example, the interval \(-5, 2\) is written as \(-5 < x < 2\). If the inequality holds for this interval, the interval notation would be \((-5, 2)\). Repeat the process for all intervals where the inequality holds.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

What is a rational function?

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+2}-2$$

Describe how to graph a rational function.
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