/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 Solve each absolute value inequa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 64

Solve each absolute value inequality. $$|3(x-1)+2| \leq 20$$

Short Answer

Expert verified
The solution to the inequality is \(-7 \leq x \leq 7\).

Step by step solution

01

Set up the inequalities

Because it is an absolute value inequality, the expression inside the absolute value can be either positive or negative. Hence, the inequality can be written as two separate inequalities: \(3(x-1)+2 \leq 20\) and \(-(3(x-1)+2) \leq 20\).
02

Solve the first inequality

Solving the first inequality involves simplifying the equation and isolating x on one side. First, distribute the 3 to get \(3x-3+2 \leq 20\). Combine like terms to get \(3x - 1 \leq 20\). Then, add 1 to both sides to get \(3x \leq 21\). Finally, divide both sides by 3, to get \(x \leq 7\).
03

Solve the second inequality

First distribute the negative sign to get \(-3x+3-2 \leq 20\). Then, combine like terms to get \(-3x+1 \leq 20\). Add 1 to both sides to get \(-3x \leq 21\). When dividing or multiplying both sides of an inequality by a negative number, the inequality symbol flips. So divide both sides by -3, to get \(x \geq -7\).
04

Combine the solutions

Both inequalities \(x \leq 7\) and \(x \geq -7\) must be true at the same time as we are considering an absolute value which is less than or equal to 20. The solution to the original inequality is thus the intersection of the solutions to each of these separate inequalities, which is \(-7 \leq x \leq 7\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain how to multiply rational expressions.

Explain how to simplify a rational expression.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When checking a radical equation's proposed solution, I can substitute into the original equation or any equation that is part of the solution process.

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.