/*! 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 30 Solve each inequality. Give the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

Solve each inequality. Give the solution set using interval notation. $$|3 x-4| \geq 2$$

Short Answer

Expert verified
The solution set is \((-\infty, \frac{2}{3}] \cup [2, \infty)\).

Step by step solution

01

Understand the Absolute Value Inequality

We need to solve the inequality \(|3x - 4| \geq 2\). Absolute value inequalities can be split into two separate inequalities: \(3x - 4 \geq 2\) or \(3x - 4 \leq -2\).
02

Solve the First Inequality

Start with the first part of the inequality: \(3x - 4 \geq 2\). Add 4 to both sides to obtain \(3x \geq 6\). Then, divide both sides by 3 to get \(x \geq 2\).
03

Solve the Second Inequality

Now solve the second part of the inequality: \(3x - 4 \leq -2\). Add 4 to both sides to get \(3x \leq 2\). Then, divide both sides by 3 to get \(x \leq \frac{2}{3}\).
04

Combine the Solution Sets

Combine the solutions of the two inequalities. Since the inequality is 'greater than or equal to,' the solution set includes all values where \(x \geq 2\) or \(x \leq \frac{2}{3}\). In interval notation, this is written as \((-\infty, \frac{2}{3}] \cup [2, \infty)\).

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.

inequality solving
Solving inequalities involves finding the range of values that satisfies the given condition. In this exercise, we are working with an absolute value inequality \( |3x - 4| \geq 2 \). Absolute value inequalities like this one can be split into two different but connected inequalities.

Here, we consider two cases:
  • The expression inside the absolute value is positive or zero:
    \( 3x - 4 \geq 2 \).

    By solving, we get:
    \( 3x \geq 6 \implies x \geq 2 \)

  • The expression inside the absolute value is negative:
    \( 3x - 4 \leq -2 \).

    By solving, we get:
    \( 3x \leq 2, \implies x \leq \frac{2}{3} \)


This means we need to consider both inequalities independently. Finally, combining these results provides the final solution set in terms of inequalities.
interval notation
Interval notation is a way to represent the solution set of an inequality in a more concise form. For our inequality \( |3x - 4| \geq 2 \), we have two sets of values:
  • Values of x where
    \( x \geq 2 \)
  • Values of x where
    \( x \leq \frac{2}{3} \)


In interval notation, these are written as:
  • \( [2, \infty) \): representing the numbers starting from 2 and going to infinity

  • \( (-\infty, \frac{2}{3}] \): representing the numbers from negative infinity up to and including \frac{2}{3}


The combined solution, using the union symbol \( \cup \), is: \( (-\infty, \frac{2}{3}] \cup [2, \infty). \)
absolute value operations
Understanding absolute value is key when solving inequalities. Absolute value represents the distance of any number from zero on the number line, regardless of direction. So, for any value x:

\( |x| = \begin{cases} x, & \text{{if }} x \geq 0 \ \ -x, & \text{{if }} x < 0 \end{cases} \)

When we say \( |3x - 4| \geq 2 \), it means:
  • Either, \( 3x - 4 \geq 2 \),
    which leads to solving \( 3x - 4 \geq 2, \text{{giving us }} x \geq 2 \)

  • Or, \( -(3x - 4) \geq 2 \) or equivalently
    \( 3x - 4 \leq -2 \), which leads to solving \( 3x - 4 \leq -2, \text{{giving us }} x \leq \frac{2}{3} \)


In both cases, we split the absolute value inequality into two separate inequalities and solve them individually. This method is essential for understanding and solving any type of absolute value inequality.

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

Buying and Selling Land Roger bought two plots of land for a total of \(\$ 120,000\). When he sold the first plot, he made a profit of \(15 \% .\) When he sold the second, he lost \(10 \% .\) His total profit was \(\$ 5500 .\) How much did he pay for each piece of land?

Explain why the equation \(|x|=\sqrt{x^{2}}\) has infinitely many solutions.

Solve each equation or inequality. $$|6-2 x|+1=3$$

Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$I=10+4 i, E=88+128 i$$

Find each quotient. Write the answer in standard form \(a+b i .\) $$ \frac{14+5 i}{3+2 i} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.