/*! 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 47 Solve the inequality. Then graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 47

Solve the inequality. Then graph the solution set. $$\frac{1}{x-3} \leq \frac{9}{4 x+3}$$

Short Answer

Expert verified
The solution to the inequality is \((- \frac{3}{4}, 3)\), where the interval includes '-3/4' and '3'

Step by step solution

01

Find Critical Values

Critical values are the numbers that make the inequality undefined or make it equal to zero. They are found by setting the denominator equal to zero.\nHence, critical values are:\(x-3 = 0\) which gives \(x = 3\) and \(4x + 3 = 0\) resulting in \(x = -\frac{3}{4}\)
02

Create Sign Chart

The identified critical values partition the number line into intervals. Substitute any number from each interval into the original inequality to determine the sign.\nThe intervals are: \( (-\infty, -\frac{3}{4}), (-\frac{3}{4}, 3), (3, +\infty) \)\nFor example, for interval \((-\infty, -\frac{3}{4})\), let's choose \(x=-1\). The result is positive.\nNext, for \((- \frac{3}{4}, 3)\), take \(x=0\). This gives a negative result.\nFor \((3, +\infty)\), let's use \(x=4\), which results to a positive output.
03

Solve Inequality Using Sign Chart

The given inequality is less than or equal to \(\leq 0\).\nWe are therefore, interested in the intervals that resulted in negative or zero. From the sign chart, the interval that satisfies the inequality is: \((- \frac{3}{4}, 3)\)
04

Graph the Solution Set

The solution to the inequality is the interval \((- \frac{3}{4}, 3)\). Draw a number line and mark the points '-3/4' and '3'. The solution is the interval between these two, and since it includes '-3/4' and '3', the interval should be closed (or filled) at these points.

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

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$

The revenue and cost equations for a product are \(R=x(50-0.0002 x)\) and \(C=12 x+150,000,\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) What is the price per unit?

Cube each complex number. (a) \(-1+\sqrt{3} i\) (b) \(-1-\sqrt{3} i\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.