/*! 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 65 Solve each absolute value inequa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 65

Solve each absolute value inequality. $$\left|\frac{2 x+6}{3}\right|<2$$

Short Answer

Expert verified
The solution to the inequality is \(x \in (-6,0)\) or all numbers that are greater than -6 but less than 0

Step by step solution

01

Setting Up the Inequalities

Since the absolute value is less than a number, we will develop two inequalities. First, we keep the inside of the absolute value as it is and then compare it with the number at the right hand side, which is 2:\[\frac{2x+6}{3}<2\]Second, we reverse the sign of the expression inside the absolute value, and also reverse the inequality sign:\[-\frac{2x+6}{3}<2\]
02

Solve the First Inequality

\[\frac{2x+6}{3}<2\]To get rid of the fraction, multiply both sides by 3:\[2x+6<6\]Subtract 6 from both sides:\[2x<0\]Finally, divide both sides by 2 to isolate x:\[x<0\]
03

Solve the Second Inequality

To solve the second inequality, the steps are almost the same, but keeping in mind of the negative sign at the start:\[-\frac{2x+6}{3}<2\]Multiply both sides by 3:\[-2x-6<6\]Add 6 to both sides:\[-2x<12\]Divide both sides by -2 (and remember when you divide or multiply an inequality by a negative number, the inequality sign flips):\[x>-6\]
04

Combine the Inequalities

Both inequalities have to be satisfied, so combine them:\[x>-6 \text{ and } x<0\]This means that x can be any number that is greater than -6 but less than 0.

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

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. The formula $$ I=\frac{1}{4} x+26 $$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$ N=\frac{1}{4} x+6 $$ models the percentage of U.S households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F\), is $$ F=\frac{9}{5} C+32 $$ If Fahrenheit temperature ranges from \(41^{\circ}\) to \(50^{\circ},\) inclusive, what is the range for Celsius temperature? Use interval notation to express this range.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and \(88 .\) There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges \(\$ 8\) per month plus 5垄 per check. The credit union charges \(\$ 2\) per month plus 8垄 per check. How many checks should be written each month to make the credit union a better deal?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

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.