/*! 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 83 Solve and graph each solution se... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 83

Solve and graph each solution set. $$ 2 \leq f(x) \leq 8, \text { where } f(x)=3 x-1 $$

Short Answer

Expert verified
The solution set is \(1 \leq x \leq 3\).

Step by step solution

01

Set Up Inequalities

We have the compound inequality: \[2 \leq f(x) \leq 8\]Substitute the given function \(f(x) = 3x - 1\) into the compound inequality: \[2 \leq 3x - 1 \leq 8\]
02

Isolate the Variable

To solve for \(x\), first add 1 to all parts of the inequality:\[2 + 1 \leq 3x - 1 + 1 \leq 8 + 1\]This simplifies to: \[3 \leq 3x \leq 9\]
03

Divide by the Coefficient

Next, divide all parts of the inequality by 3 to isolate \(x\):\[\frac{3}{3} \leq \frac{3x}{3} \leq \frac{9}{3}\]This simplifies to: \[1 \leq x \leq 3\]
04

Graph the Solution Set

To graph the solution set, plot points at \(x = 1\) and \(x = 3\) on a number line, including these endpoints as they are part of the solution. Then shade the interval between 1 and 3.

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
When solving inequalities, our main goal is to find the set of values that make the inequality true. For instance, in the compound inequality \( 2 \leq f(x) \leq 8 \), we aim to isolate the variable to understand which values of \( x \) satisfy this condition. This involves following a set of steps: substituting the function, simplifying, and carefully manipulating the inequality while respecting its rules. We must keep in mind that inequalities flip when we multiply or divide by a negative number. This nuance is crucial to ensure our solution is accurate.
inequality graphing
Graphing inequalities visually represents the solution set on a number line or coordinate plane. For the inequality \( 1 \leq x \leq 3 \), we mark the endpoints. In this case, we include 1 and 3, indicated by closed dots, because the inequality signs \( \leq \) mean these values are part of the solution. Connect the dots and shade the space in between, representing all numbers from 1 to 3. Graphs simplify understanding of the solution range. For complex inequalities, graphing provides a clear method to visualize valid values.
isolate variable
Isolating the variable involves manipulating the inequality to get the variable alone on one side. Begin with the simplest operation, such as addition or subtraction, which affects all parts of the inequality equally. As shown, adding 1 to each part of \( 2 \leq 3x - 1 \leq 8 \) results in \( 3 \leq 3x \leq 9 \). Next, divide by the coefficient of the variable, like the 3 in \( 3x \). Maintain balance by dividing each part of the inequality, giving us \( 1 \leq x \leq 3 \). This method ensures we systematically approach the solution, avoiding errors.

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

For \(g(x)=\frac{2+3 x}{2 x-4},\) find all \(x\) -values for which \(g(x) \geq 0\).

For \(f(x)\) as given, use interval notation to write the domain of \(f\) $$ f(x)=\sqrt{8-2 x} $$

To prepare for Chapter \(9,\) review solving systems of equations using elimination (Section 4.3). Solve. [ 4.3] $$ \begin{aligned} &y-5 x=2\\\ &3 x+y=-1 \end{aligned} $$

Solve using substitution or elimination. $$ \begin{array}{r} {x-3 y=8} \\ {2 x+3 y=4} \end{array} $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\) $$ f(x)=\frac{2}{x+3} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.