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Chapter 2: Problem 54

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{(x+3)(x-2)}{x+1} \leq 0$$

Short Answer

Expert verified
The solution set in interval notation is \((-\infty, -3] \cup [-1, 2)\)

Step by step solution

01

Factor out the equation

The given equation is already factored out, so we do not need to do any additional work in this step. The equation is: \((x+3)(x-2)/(x+1) \leq 0\)
02

Find critical points

The critical points for this equation are the zeros of the numerator and the values which make the denominator undefined. Therefore, setting each piece of the rational inequality equal to 0 for \((x+3)(x-2) = 0\) we get \(x = -3, 2\). Setting the denominator equal to 0 for \(x+1 = 0\), we get \(x = -1\). So, the critical points are \(x = -3, 2, -1\).
03

Test values from each interval

Let's test some values from each interval: Choose \(x=-4\) for the interval \((-\infty, -3)\), \(x=-2\) for the interval \((-3, -1)\), \(x=0\) for the interval \((-1, 2)\), and \(x=3\) for the interval \((2, \infty)\).Substitute these values in the given rational expression. This we will give a better understanding whether the intervals will result in positive or negative values.
04

Write the solution set in interval notation

After calculating a value for each interval, it will be determined that the intervals \((-3, -1)\) and \((2, \infty)\) result in positive values, so they are excluded. The solution set in interval notation is \((-\infty, -3] \cup [-1, 2)\)

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Most popular questions from this chapter

You drive from your home to a vacation resort 600 miles away. You return on the same highway. The average velocity on the return trip is 10 miles per hour slower than the average velocity on the outgoing trip. Express the total time required to complete the round trip, \(T\), as a function of the average velocity on the outgoing trip, \(x .\)

Find the horizontal asymptote, if there is one, of the graph of rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x-\frac{1}{x}}{x+\frac{1}{x}}$$

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)
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