Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.7 - Polynomial and Rational Inequalities - Exercise Set - Page 413: 53


$(-∞, -4] ∪ (-2, 1]$

Work Step by Step

Consider the Rational Inequality as follows: $\frac{(x+4)(x-1)}{x+2}≤0$ Here are the steps required for Solving Rational Inequalities: Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. $\frac{(x+4)(x-1)}{x+2}≤0$ Step 2: Critical or Key Values are first evaluated. In order to this, set the numerator and denominator of the fraction equal to zero and solve. $x+4 = 0$ This implies $x =-4$ and $x-1=0$ This implies $ x =1$ Also, $x+2=0$ This implies $ x =-2$ These solutions are used as boundary points on a number line. Step 3: Locate the boundary points on a number line and divide the number line into intervals. The boundary points divide the number line into four intervals: $(-∞, -4), (-4, -2),(-2, 1), (1, ∞)$ Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number. Intervals: $(-∞, -4), (-4, -2), (-2, 1), (1, ∞)$ Test value: $-5$ $-3$ $0$ $2$ Sign Change: Negative Positive Negative Positive $f (x)< 0?$: T F T F Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x)≤ 0$. Based on our work done in Step 4, we see that $f (x)≤ 0$ for all x in $(-∞, -4]$ or $(-2, 1]$ . However, the inequality involves (less than or equal to), we must also include the solution of $f (x)= 0$ , namely -4 and 1 in the solution set. Conclusion: Thus, the interval notation of the given inequality is $(-∞, -4] ∪ (-2, 1]$ and the graph of the solution set on a number line is shown as follows:
Small 1518097249
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.