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Chapter 2: Problem 48

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

Short Answer

Expert verified
The solution set of the inequality is \([-3, -2) \cup (-2, 鈭�)\].

Step by step solution

01

Identifying the Critical Values

Set both the numerator and the denominator of the inequality to zero and solve each equation separately for the variable. \(-x - 3 = 0\) gives \(x = -3\). \(x + 2 = 0\) gives \(x = -2\). However, \(x\) cannot equal \(-2\) because that would make the denominator equal to zero, causing the rational expression to be undefined.
02

Interval Checking

The number line is divided into three intervals by the critical values: \((-鈭�, -3)\), \((-3, -2)\), and \((-2, 鈭�)\). Choose a test point in each interval. For example, for the interval \((-鈭�, -3)\), choose \(x=-4\); for \((-3, -2)\), choose \(x=-2.5\); and for \((-2, 鈭�)\), choose \(x=0\). Substitute these values into the original inequality.
03

Solving the Inequality

When \(x = -4\), the left side of the expression becomes \(\frac{-(-4)-3}{-4+2} = -0.5\), which is greater than zero. Therefore the interval \((-鈭�, -3)\) does not satisfy the inequality. When \(x = -2.5\), the left side becomes \(\frac{-(-2.5)-3}{-2.5+2} = 6\), which is still greater than zero. Hence, the interval \((-3, -2)\) doesn't satisfy the inequality. When \(x=0\), the expression is \(\frac{-0-3}{0+2} = -1.5\), which is less than zero. Thus, the interval \((-2, 鈭�)\) satisfies the inequality. Additionally, since the inequality is \(\leq 0\), \(-3\) is also included in the solution set because it makes the left side exactly equal to 0.
04

Writing the Solution in Interval Notation

From the above results, the solution set will be \([-3, -2) \cup (-2, 鈭�)\].

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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 of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{2 x-9}{x-4}$$

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if any, of the function's graph.

Find the inverse of \(f(x)=x^{3}+2\)

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).
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