/*! 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 53 Solve each rational inequality a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 53

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{(x+4)(x-1)}{x+2} \leq 0$$

Short Answer

Expert verified
The solution of the rational inequality \(\frac{(x+4)(x-1)}{x+2} \leq 0\) on a real number line and expressed in interval notation is \((-鈭�, -4] \cup [-2, 1]\).

Step by step solution

01

Identify Potential Solutions

Firstly, isolate the inequality. Find the values of \(x\) that make the rational expression equal to zero. This can be done by setting the numerator equal to zero since anything multiplied by zero results in zero: \((x+4)(x-1) = 0\). This gives us \(x = -4, 1\). These are the potential solutions unless they make the denominator zero.
02

Check for Undefined Values

The denominator must not be zero because division by zero is undefined in mathematics. Thus, determine if any of the potential solutions would make the denominator equal to zero by solving the equation \(x+2 = 0\). This gives us \(x = -2\), which is excluded from the solution set because it would make the denominator zero.
03

Dividing the Number Line

Now using the obtained values, divide the real number line into intervals and choose a test point from each interval to decide about the sign of that interval i.e., whether the expression becomes positive or negative. The three critical numbers are \(x=-4\), \(x=-2\), and \(x=1\). This divides the number line into four intervals: \((-鈭�, -4)\), \((-4, -2)\), \((-2, 1)\), \( (1, 鈭�)\). Choose a test value from each interval such as \(x=-5\), \(x=-3\), \(x=0\), and \(x=2\) respectively.
04

Test the Sign of Each Interval

Substitute each test value into the inequality to determine the sign of the expression within that interval. If the expression is positive for a particular test value, it means it is positive for the entire interval. If it is negative for a certain test value, it is negative for the entire interval. After testing each interval, the signs are found to be -, +, -, + respectively.
05

Find the Solution Set and Express in Interval Notation

The inequality \(\leq 0\) includes zero, so we include the values of \(x\) where the expression equals zero (unless they caused division by zero). So the solution to the inequality is when the expression is less than or equal to zero. From the sign pattern, it can be seen that the solution set is \((-鈭�, -4] \cup [-2, 1]\).

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 transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x}+2$$

Describe how to graph a rational function.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-4$$

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

Find the horizontal asymptote, if there is one, of the graph of rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Probability and 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.