/*! 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 23 Solve each polynomial inequality... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 23

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$-x^{2}+x \geq 0$$

Short Answer

Expert verified
The solution set is \(x \in [0,1]\).

Step by step solution

01

Rearrange the Inequality

Rearrange the inequality as \(x^{2} - x \leq 0\). We can write this in the form \(x(x-1) \leq 0\).
02

Find the Critical Points

Find the values of \(x\) for which \(x(x-1) = 0\). It is clear to see that the critical points are \(x = 0\) and \(x = 1\).
03

Find the Intervals

Separate the real number line into intervals based on the critical points: \((- \infty, 0)\), \((0, 1)\), and \((1, \infty)\). Check each interval to see if it satisfies the inequality.
04

Check the Interval (-鈭�, 0)

Substitute a number from the interval \((- \infty, 0)\), for example \(-1\), into the inequality. We have \((-1)(-1-1) = -2\), which is not less than or equal to zero, hence this interval is not part of the solution.
05

Check the Interval (0, 1)

Substitute a number from the interval \((0, 1)\), for example \(0.5\), into the inequality. We have \((0.5)(0.5-1) = -0.25\), which is less than or equal to zero, hence this interval is part of the solution.
06

Check the Interval (1, 鈭�)

Substitute a number from the interval \((1, \infty)\), for example \(2\), into the inequality. We have \((2)(2-1) = 2\), which is not less than or equal to zero, hence this interval is not part of the solution.
07

Graph the Solution

Since only the interval (0,1) satisfies the inequality, this is the solution set. On a number line, this would be displayed as a filled circle at 0, a line connecting 0 and 1, and a filled circle at 1 (to indicate the interval is inclusive).
08

Express in Interval Notation

Express the solution set in interval notation. The solution in interval notation is \([0,1]\).

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. $$h(x)=\frac{1}{x^{2}}-3$$

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. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

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. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of this direct variation equation that has a positive constant of variation shows one variable increasing as the other variable decreases.

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 of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x-7}{x-2}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.