/*! 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 97 When 3 times a number is subtrac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 97

When 3 times a number is subtracted from \(4,\) the absolute value of the difference is at least \(5 .\) Use interval notation to express the set of all numbers that satisfy this condition.

Short Answer

Expert verified
The numbers that satisfy the given condition are \([- \infty, -1/3] \cup [3, + \infty]\).

Step by step solution

01

- Set up the inequality

Start the problem by setting up the inequality according to the instructions given in the problem. The difference of 4 and three times a number is at least 5. This can be written as \( |4 - 3x| \geq 5 . \)
02

- Solve the positive case

In this step, solve the equation assuming that the inside of the absolute value (4 - 3x) is positive. This means \(4 - 3x \geq 5\). To solve for x, begin by subtracting 4 from both sides, resulting in \(-3x \geq 1\). Then, divide both sides by -3 (keep in mind that when you divide or multiply an inequality by a negative number, you have to flip the inequality). So, the x value for the positive case is \(x \leq -1/3 .\)
03

- Solve the negative case

Now solve the equation assuming that the inside of the absolute value (4 - 3x) is negative. This means \(4 - 3x \leq -5\). To solve for x, begin by subtracting 4 from both sides, resulting in \(-3x \leq -9\). Then, divide both sides by -3 (keeping in mind the sign flipping rule). So, the x value for the negative case is \(x \geq 3 .\)
04

- Express the solution in interval notation

The solution to the problem is the union of the solutions of the two cases. In interval notation, the solution to this inequality would be \([- \infty, -1/3] \cup [3, + \infty]\). This means that any number less than or equal to -1/3 or greater than or equal to 3 will satisfy the original inequality.

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

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.