/*! 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 34 Solve each polynomial inequality... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 34

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (5-x)^{2}\left(x-\frac{13}{2}\right)<0 $$

Short Answer

Expert verified
The solution to the polynomial inequality \((5-x)^{2}\left(x-\frac{13}{2}\right)<0\) in interval notation is \((- \infty, 5) \cup (6.5, \infty)\).

Step by step solution

01

Factor Polynomial

The polynomial is already factored in the exercise which is \((5-x)^{2}\left(x-\frac{13}{2}\right) < 0\)
02

Identify Critical Points

The polynomial will be zero at its critical points. These points are the solutions to the equation \((5-x)^{2}\left(x-\frac{13}{2}\right) = 0\). This gives the solutions \(x=5\) (from \(5-x=0\)) and \(x=\frac{13}{2}\) (from \(x-\frac{13}{2}=0\)). Thus, the critical points are \(5\) and \(\frac{13}{2}\) or \(6.5\).
03

Test the Intervals

The critical points divide the number line into 3 regions: \(-\infty\) to \(5\), \(5\) to \(\frac{13}{2}\ = 6.5\), and \(6.5\) to \(\infty\). Pick a number from each of these intervals and substitute into the inequality. Choose \(x=4\), \(x=6\) and \(x=7\) for these tests. For \(x=4\), the inequality will be \((5-4)^2 * (4-\frac{13}{2}) > 0\), which is positive. For \(x=6\), the inequality will be \((5-6)^2 * (6-\frac{13}{2}) < 0\), which is negative. For \(x=7\), the inequality will be \((5-7)^2 * (7-\frac{13}{2}) > 0\), which is positive. Hence, the inequality holds true for \(-\infty < x < 5\) and \(6.5 < x < \infty\)
04

Write in Interval Notation

The solution in interval notation is \(-\infty < x < 5\) and \(6.5 < x < \infty\). This interval can be written as \((- \infty, 5) \cup (6.5, \infty)\)

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

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

Will help you prepare for the material covered in the next section. $$ \text { Solve: } 2 x^{2}+x=15 $$

If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and falls to the right.

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=x^{3}+13 x^{2}+10 x-4$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.