/*! 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 43 Solve each rational inequality i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 43

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x-4}{x+3}>0 $$

Short Answer

Expert verified
The solution to the given inequality is \(x \in (-3,4)\)

Step by step solution

01

Identify Critical Points

Critical points are the values that either makes the inequality 0 or undefined. Roots of the numerator, as well as the values that make denominator 0 are the critical points. For \(\frac{x-4}{x+3}>0\), the critical points are \(x=4\) (root of the numerator) and \(x=-3\) (which makes denominator equal to 0).
02

Define the Intervals

Form intervals based on critical points in ascending order. Here, our critical points divide the number line into three intervals: \(-\infty,-3\), \(-3,4\), and \(4,\infty\).
03

Test Each Interval

For an interval, take a test point and substitute it in the inequality. Note that an interval may satisfy the inequality, not satisfy the inequality or undefined. Testing these intervals we get: \nFor \(-\infty,-3\), test point is \(-4\), \(\frac{-4-4}{-4+3}<0\), which is FALSE. \nFor \(-3,4\), test point is 0, \(\frac{0-4}{0+3}<0\), which is TRUE. \nFor \(4,\infty\), test point is 5, \(\frac{5-4}{5+3}>0\), which is FALSE.
04

Interpret the Results

Since we are looking for \(\frac{x-4}{x+3}>0\), and the interval \(-3'.
05

Express the Solution in Interval Notation and Plot

The solution to the inequality in interval notation is \((-3,4)\). To plot this on a number line, mark an open circle at -3 and 4, and shade the region between them indicating all the numbers between -3 and 4 are the solutions to the given 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

What are the zeros of a polynomial function and how are they found?

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has a vertical asymptote given by x=3, a horizontal asymptote y=0, y -intercept at -1, and no x -intercept.

Use $$\frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2$$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3} $$

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.