College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 4 - Section 4.5 - Inequalities involving Quadratic Functions - 4.5 Assess Your Understanding - Page 314: 13

Answer

The inequality is valid for values less than -4 and more than 3 (not including them) i.e. $(-\infty,-4)\cap (3,\infty)$

Work Step by Step

First, we are going to set the right side to zero and factor to find the x-intercepts: $x^2+x-12=0$ $(x+4)(x-3)=0$ $x_1=-4$ $x_2=3$ These are the critical points. We are going to take three values: one less than -4, one between -4 and 3, and one more than 3 to test in the original equation and check if the inequality is true or not: First test with a value less than -4: $(-5)^2+(-5)>12$ $25-4>12$ $21>12 \rightarrow \text{ TRUE}$ Second test with a value between -4 and 3: $(0)^2+0>12$ $0+0>12$ $0>12 \rightarrow \text{ FALSE}$ Third test with a value more than 3: $(4)^2+4>12$ $16+4>12$ $20>12 \rightarrow \text{ TRUE}$ These tests show that the inequality $x^2+x>12$ is valid for values less than -4 and more than 3 (not including them) i.e. $(-\infty,-4)\cap (3,\infty)$
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