College Algebra (10th Edition)

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

Chapter 5 - Section 5.4 - Polynomial and Rational Inequalities - 5.4 Assess Your Understanding - Page 373: 32

Answer

The inequality is valid for values more than 1 (not including it) i.e. $ (1,\infty)$.

Work Step by Step

First, we are going to find the x-intercepts by equating to zero: $x^3=1$ $x^3-1=0$ $(x-1)(x^2+x+1)=0$ The expresion $x^2+x+1$ cannot equal zero, so that leaves us with the only x-intercept: $(x-1)=0$ $x=1$ This is the only critical point. We are going to take two values: one less than 1 and one more than 1 to test in the original equation and check if the inequality is true or not: First test with a value less than 1: $(0)^3>1$ $0>1 \rightarrow \text{ FALSE}$ Second test with a value more than 1: $2^4>1$ $16>1 \rightarrow \text{ TRUE}$ These tests show that the inequality $x^3>1$ is valid for values more than 1 (not including it) i.e. $ (1,\infty)$
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