/*! 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 79 Solve and graph each solution se... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 79

Solve and graph each solution set. $$ 5>\frac{x-3}{4}>1 $$

Short Answer

Expert verified
The solution set is \( 7 < x < 23 \) and can be graphed with open circles at 7 and 23, shading the region between them.

Step by step solution

01

- Isolate the variable

The given compound inequality is \[ 5 > \frac{x-3}{4} > 1 \] This compound inequality can be broken into two separate inequalities: \[ 5 > \frac{x-3}{4} \] and \[ \frac{x-3}{4} > 1 \]
02

- Solve the first inequality

To solve the first inequality \( 5 > \frac{x-3}{4} \), multiply both sides of the inequality by 4 to eliminate the fraction: \[ 5 \times 4 > \frac{x-3}{4} \times 4 \] This simplifies to: \[ 20 > x - 3 \] Add 3 to both sides to isolate x: \[ 20 + 3 > x \] Therefore, \[ 23 > x \]
03

- Solve the second inequality

To solve the second inequality \( \frac{x-3}{4} > 1 \), multiply both sides of the inequality by 4: \[ \frac{x-3}{4} \times 4 > 1 \times 4 \] This simplifies to: \[ x - 3 > 4 \] Add 3 to both sides to isolate x: \[ x - 3 + 3 > 4 + 3 \] Therefore, \[ x > 7 \]
04

- Combine the solutions

Combine the results from Step 2 and Step 3. The solution set of the compound inequality is: \[ 7 < x < 23 \]
05

- Graph the solution set

To graph the solution set \( 7 < x < 23 \), draw a number line, place open circles on 7 and 23, and shade the region between them to indicate that x can be any value between 7 and 23, but not including 7 and 23.

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.

solving inequalities
Solving inequalities involves finding the range of values for a variable that makes the inequality true. In this problem, we start with the compound inequality: \( 5 > \frac{x-3}{4} > 1 \). A compound inequality means that the variable must satisfy two inequalities simultaneously. First, we break it into separate inequalities: \( 5 > \frac{x-3}{4} \) and \( \frac{x-3}{4} > 1 \). Next, we solve each part by isolating the variable, as we'll see in the 'variable isolation' section. This step-by-step approach helps to methodically find all possible values for x that maintain the inequality.
variable isolation
Isolating the variable means getting the variable alone on one side of the inequality. We do this to find the value range for x. Take \( 5 > \frac{x-3}{4} \):
- Multiply both sides by 4 to clear the fraction: \( 5 \times 4 > \frac{x-3}{4} \times 4 \)
- Simplify: \( 20 > x-3 \)
- Add 3 to both sides to isolate x: \( 20 + 3 > x \) or \( 23 > x \).
The process is similar for the second part:
- Multiply both sides by 4: \( \frac{x-3}{4} \times 4 > 1 \times 4 \)
- Simplify: \( x-3 > 4 \)
- Add 3 to both sides: \( x-3+3 > 4+3 \) or \( x > 7 \).
Combining both gives us: \( 7 < x < 23 \. \) Keeping these steps clear helps in accurate calculations when dealing with inequalities.
graphing inequalities
Graphing inequalities visually shows the range of possible solutions on a number line. For \(7 < x < 23\):
- Draw a horizontal number line.
- Place open circles at 7 and 23 to indicate these values are not included.
- Shade the region between the circles to show all values x can take.
This visual aid helps you and others understand the variable's allowed range quickly. Using open circles and a shaded area accurately represents that x can be any number between 7 and 23, but not equal to 7 or 23. Graphs are invaluable tools for checking your work and for clear communication of inequality solutions.

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 a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$ f(x)=x+\frac{1}{x} $$

Solve by graphing. $$ x+4=3 $$

For \(g(x)=(x-2)(x-3)(x+1),\) find all \(x\) -values for which \(g(x)>0\).

For \(f(x)\) as given, use interval notation to write the domain of \(f\) $$ f(x)=\sqrt{8-5 x} $$

Solve and graph. $$ x-10<5 x+6 \leq x+10 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and 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.