/*! 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 13 Solve each polynomial inequality... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 13

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. $$ 2 x^{2}+x<15 $$

Short Answer

Expert verified
The solution is \( x \in (-\infty, -3) \cup (2.5, \infty) \).

Step by step solution

01

Rearrange the inequality

Rearrange the inequality into a standard form, like \( ax^2+bx+c<0 \) or \( ax^2+bx+c>0 \). This is done by subtracting 15 from both sides resulting in \( 2x^2 + x - 15 < 0 \).
02

Factor the quadratic

To find the roots, factoring is generally the best approach. Factors of \( 2x^2 + x - 15 \) are \( (2x - 5)(x + 3) \). So, the inequality becomes \( (2x-5) (x+3) < 0 \).
03

Find the roots

Set each factor equal to 0 and solve for \(x\). The roots are \(x = 2.5\) and \(x = -3\). These are the values of \(x\) that make the inequality equal to 0.
04

Perform a sign analysis

Consider the intervals formed by the roots on the number line (-infinity,-3), (-3,2.5), (2.5,infinity). Choose a test number from each interval and evaluate the inequality. The intervals (-infinity,-3) and (2.5,infinity) make the inequality truthful.
05

Interpret the solution in interval notation

The solution set in interval notation, based on the sign analysis, which makes the inequality true is (-infinity,-3) U (2.5,infinity). Be aware that interval notation uses round brackets for 'less than' or 'greater than', and square brackets for 'less than or equal to' or 'greater than or equal to'.

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

Although I have not yet learned techniques for finding the \(x\) -intercepts of \(f(x)=x^{3}+2 x^{2}-5 x 6,\) I can easily determine the \(y\) -intercept.

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. $$f(x)=x^{3}(x+2)^{2}(x+1)$$

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. $$f(x)=-2 x^{3}(x-1)^{2}(x+5)$$

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. $$f(x)=-x^{2}(x-1)(x+3)$$

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. $$f(x)=6 x^{3}-9 x-x^{5}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

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.