Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 1 - Equations and Inequalities - 1.8 Absolute Value Equations and Inequalities - 1.8 Exercises: 52

Answer

The solution is $(-\infty,1]\cup[3,\infty)$

Work Step by Step

$|12-6x|+3\ge9$ Take $3$ to the right side of the inequality: $|12-6x|\ge9-3$ $|12-6x|\ge6$ Solving this absolute value inequality is equivalent to solving two separate inequalities, which are: $12-6x\ge6$ and $12-6x\le-6$ $\textbf{Solve the first inequality:}$ $12-6x\ge6$ Take $12$ to the right side: $-6x\ge6-12$ $-6x\ge-6$ Take $-6$ to divide the right side and reverse the direction of the inequality sign: $x\le\dfrac{-6}{-6}$ $x\le1$ $\textbf{Solve the second inequality:}$ $12-6x\le-6$ Take $12$ to the right side: $-6x\le-6-12$ $-6x\le-18$ Take $-6$ to divide the right side and reverse the direction of the inequality sign: $x\ge\dfrac{-18}{-6}$ $x\ge3$ Expressing the solution in interval notation: $(-\infty,1]\cup[3,\infty)$
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