/*! 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 29 Solve each inequality. Give the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

Solve each inequality. Give the solution set using interval notation. $$|2 x+5| \geq 3$$

Short Answer

Expert verified
The solution set in interval notation is \ \left( -\infty, -4 \right] \cup \left[ -1, \infty \right) \.

Step by step solution

01

Understand the absolute value inequality

The given inequality is \( |2x + 5| \geq 3 \). Absolute value inequalities of the form \( |A| \geq B \) can be split into two separate inequalities: \( A \geq B \) or \( A \leq -B \).
02

Split the inequality

Split the inequality into two parts: \( 2x + 5 \geq 3 \) or \( 2x + 5 \leq -3 \).
03

Solve the first inequality

Solve \( 2x + 5 \geq 3 \): \ 2x \geq -2 \, then divide by 2 to get \ x \geq -1 \.
04

Solve the second inequality

Solve \( 2x + 5 \leq -3 \): \ 2x \leq -8 \, then divide by 2 to get \ x \leq -4 \.
05

Combine the solution sets

The solution set is the combination of both inequalities: \ x \leq -4 \) or \( x \geq -1 \). In interval notation, this is written as \( \left( -\infty, -4 \right] \cup \left[ -1, \infty \right) \.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

interval notation
Interval notation is a concise way of representing sets of numbers, often used to describe solution sets for inequalities.
Instead of listing all the values within a range, interval notation uses parentheses and brackets to denote the start and end points of an interval.
Here are some key points:
  • Parentheses ( ) indicate that the endpoint is not included (open interval).
  • Brackets [ ] indicate that the endpoint is included (closed interval).
  • A combination like (-鈭�, 5] means all values less than or equal to 5.
  • Union (鈭�) is used to combine multiple intervals, indicating that the solution set includes values from either interval.

Remembering these rules can help you accurately express the solutions to inequalities using interval notation.
splitting inequalities
When dealing with absolute value inequalities like |2x + 5| 鈮� 3, the process of 'splitting' the inequality is crucial.
Absolute value represents the distance from zero, so when it is greater than or equal to a number, the expression inside can be either positive or negative.
Here's how you can split it:
  • First, understand the form: |A| 鈮� B translates to A 鈮� B or A 鈮� -B.
  • In our example, |2x + 5| 鈮� 3 gets split as 2x + 5 鈮� 3 or 2x + 5 鈮� -3.
  • Solving these two separate inequalities provides the ranges where the original inequality holds true.

Splitting the inequalities in this way enables us to identify distinct solution sets, leading to a comprehensive and correct answer.
solution sets
Solution sets represent the values that satisfy a given inequality or equation.
At the end of solving an absolute value inequality, you combine the results from the 'splitting' process to form the final solution set.
Here's how to interpret solution sets:
  • After splitting |2x + 5| 鈮� 3, we found x 鈮� -1 or x 鈮� -4.
  • The solution set is all values that make at least one of these inequalities true.
  • In interval notation, this is written as (-鈭�, -4] 鈭� [-1, 鈭�), meaning values less than or equal to -4 or greater than or equal to -1.

Understanding solution sets and expressing them correctly in interval notation ensures clarity and accuracy in mathematical communication.

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

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{12}{-i}$$

Velocity of an Object The velocity of an object, \(v,\) after \(t\) seconds is given by $$ v=3 t^{2}-18 t+24 $$ Find the interval where the velocity is negative.

A local group of scouts has been collecting old aluminum cans for recycling. The group has already collected \(12,000\) Ib of cans, for which they could currently receive \(\$ 4\) per hundred pounds. The group can continue to collect cans at the rate of 400 lb per day. However, a glut in the old-can market has caused the recycling company to announce that it will lower its price, starting immediately, by \(\$ 0.10\) per hundred pounds per day. The scouts can make only one trip to the recycling center. How many days should they wait in order to get \(\$ 490\) for their cans?

A charter bus company charges a fare of \(\$ 40\) per person, plus \(\$ 2\) per person for each unsold seat on the bus. If the bus holds 100 passengers and \(x\) represents the number of unsold seats, how many passengers must ride the bus to produce revenue of \(\$ 5950 ?\) ( Note: Because of the company's commitment to efficient fuel use, the charter will not run unless filled to at least half-capacity.)

Solve each equation or inequality. $$\left|2 x+\frac{1}{3}\right|+1<4$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.