/*! 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 52 Solve the inequality and write t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 52

Solve the inequality and write the solution set in interval notation. \(4 x^{3}-x^{4} \geq 0\)

Short Answer

Expert verified
The solution to the inequality \(4 x^{3}-x^{4} \geq 0\) in interval notation is (0, 4] 鈭� [4, 鈭�).

Step by step solution

01

Factorise the Polynomial

The first step is to factorise the given polynomial. In this inequality, it can be factorised by factoring out the greatest common factor, \(x^{3}\), from each term. This results in \(x^{3}(4-x) \geq 0\)
02

Determine Critical Points

Critical points are solutions when the inequality equals 0. By setting \(x^{3}\) and \(4-x\) to zero, the critical values are 0 and 4.
03

Test Intervals

Create intervals using the critical numbers which will then be tested in the inequality. The intervals are (-鈭�, 0), (0, 4), (4, 鈭�). Pick a test number from each interval and substitute into the inequality. For (-鈭�, 0), a value of -1 can be chosen, for (0, 4), a value of 1, and for (4, 鈭�), a value of 5. After testing these values, the inequality holds true for intervals (0, 4) and (4, 鈭�) but does not hold true for interval (-鈭�, 0)
04

Write Answer in Interval Notation

After obtaining the intervals where the inequality holds true, write the answers in interval notation as a final answer. For the given inequality, the solution set will be in the form (0, 4] 鈭� [4, 鈭�).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
Polynomial factorization is a key technique in solving inequalities. It involves breaking a polynomial down into simpler parts, or factors, which can make the inequality easier to solve. For example, given the polynomial inequality \(4x^3 - x^4 \geq 0\), the first step is to factor the polynomial. This is done by identifying the greatest common factor (GCF) that can be extracted from each term of the polynomial. In this case, \(x^3\) is the GCF.

By factoring out \(x^3\), the polynomial simplifies to \(x^3(4-x) \geq 0\). Now, we have the polynomial in a factored form, which is simpler to work with when determining solutions.
  • Benefits of Factoring: Easier identification of critical points.
  • Simplifies Polynomial: Offers a clearer view of possible values.
Critical Points
Critical points are crucial in solving inequalities as they represent the values where the inequality is equal to zero or undefined. These points help break down the number line into intervals that can be tested. For the inequality \(x^3(4-x) \geq 0\), finding the critical points involves setting each factor to zero:
  • Set \(x^3 = 0\), giving the critical point \(x = 0\).
  • Set \(4-x = 0\), solving for \(x = 4\).
These critical points divide the number line into distinct intervals: \((-鈭�, 0)\), \((0, 4)\), and \((4, 鈭�)\). Testing values from each interval in the inequality can determine where the solution is valid.
  • Purpose of Critical Points: Help in interval testing.
  • Significance: Indicate changes in inequality direction.
Interval Notation
Interval notation is a concise way to express the solution set of an inequality. It uses brackets and parentheses to describe where an inequality holds true on a number line.

Once testing is done across intervals defined by critical points, we identify which intervals satisfy the inequality. Using the critical points \(x=0\) and \(x=4\), we test these intervals to determine the solution set:
  • The inequality holds for intervals \((0, 4)\) and \((4, 鈭�)\).
  • It fails in the interval \((-鈭�, 0)\).
Therefore, the solution in interval notation becomes \((0, 4]\cup[4, 鈭�)\), where the brackets indicate whether values are included or excluded.
  • Round Brackets \(( )\): Indicates values are not included.
  • Square Brackets \([ ]\): Indicates values are included.

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

Consider the domains of the expressions \(\sqrt[3]{x^{2}-7 x+12}\) and \(\sqrt{x^{2}-7 x+12}\). Explain why the domain of \(\sqrt{x^{2}-7 x+12}\) is different from the domain of \(\sqrt[3]{x^{2}-7 x+12}\)

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). An equation that models the data you obtained is \(y=1.4 x-39\) (a) Estimate the values of \(x\) that predict a maximum bench-press weight of at least 200 pounds. (b) Do you think an athlete's weight is a good indicator of the athlete's maximum bench-press weight? What other factors might influence an individual's bench-press weight?

Physicians consider an adult's body temperature \(x\) (in degrees Fahrenheit) to be normal if it satisfies the inequality \(|x-98.6| \leq 1\) Determine the range of temperatures that are considered to be normal.

Find the test intervals of the inequality. \(\frac{x-3}{x-1}<2\)

Solve the inequality. Then graph the solution set on the real number line. \(3|4-5 x| \leq 9\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.