/*! 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 69 Solve each inequality in Exercis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 69

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{x^{2}-x-2}{x^{2}-4 x+3}>0 $$

Short Answer

Expert verified
The solution to the inequality is \(x < -1\) or \(x>2\).

Step by step solution

01

Simplify the Inequality

First, simplify the inequality by factoring the numerator and the denominator.So, \(\frac{x^{2}-x-2}{x^{2}-4 x+3}>\) can be written as \(\frac{(x-2)(x+1)}{(x-1)(x-3)}>0\)
02

Find the Critical Points

Critical points are the values of \(x\) at which the given equation equals zero or is undefined. For \(\frac{(x-2)(x+1)}{(x-1)(x-3)}=0\), we get \(x=2, -1\). The values of \(x\) where the equation is undefined are when \(x=1, 3\). So, the critical points are \(x=-1, 1, 2, 3\).
03

Check the Intervals

Now, we need to test the sign of the expression on intervals determined by these critical points. Choose a test point from each interval and substitute that in the inequality.\nThe intervals and their signs will be: \nFor \(- \infty < x < -1\), choose \(x=-2\), the sign is +; \nfor \(-1 < x < 1\), choose \(x=0\), the sign is -; \nfor \(1 < x < 2\), choose \(x=1.5\), the sign is -; \nfor \(2 < x < 3\), choose \(x=2.5\), the sign is +, \nand for \(3 < x < \infty\), choose \(x=4\), the sign is +.
04

Drawing the Solution on the Number Line

Mark the critical points \(x=-1, 1, 2, 3\) on the number line. The inequality is \(>\) 0. So we are looking for the intervals where the sign is positive. These are the intervals \(- \infty < x < -1\) and \(2 < x < \infty\). Shade these intervals on the number line. However, since there is no 鈥渆qual to鈥� in the inequality, we put open circles at the critical 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

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{2 x-9}{x-4} $$

There is more than one third-degree polynomial function with the same three x-intercepts.

A mong all deaths from a particular disease, the percentage that is smoking related \((21-39\) cigarethes per day) is a function of the discase's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups. For example, the incidence ratio of 9 in the table means that smokers befween the ages of 65 and 74 are 9 times more likely than nonsmokers in the same age group to die from lung cancer. The rational function$$ P(x)=\frac{100(x-1)}{x} $$models the percentage of smoking-related deaths among all deaths from a disease, \(P(x),\) in terms of the disease's incidence ratio, \(x\). The graph of the rational function is shown. What is the horizontal asymptote of the graph? Describe what this means about the percentage of deaths caused by smoking with increasing incidence ratios.

In Exercises \(98-99,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}$$

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=1, a slant asymptote whose equation is y=x, y -intercept at 2, and x -intercepts at -1 and 2.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics 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.