/*! 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 59 For what positive numbers is the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 59

For what positive numbers is the cube of the number greater than four times its square?

Short Answer

Expert verified
The positive numbers greater than 4.

Step by step solution

01

Define the Inequality

Let's denote the number as x (where x is positive). We need to find when the cube of the number is greater than four times its square. This can be written as an inequality: \[x^3 > 4x^2\]
02

Simplify the Inequality

To solve the inequality, first, divide both sides by \(x^2\). Since x is positive, we can safely divide without changing the inequality direction: \[\frac{x^3}{x^2} > \frac{4x^2}{x^2}\] This simplifies to: \[x > 4\]
03

Identify the Solution Set

The inequality \(x > 4\) tells us that the positive numbers satisfying the given condition are those greater than 4.

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.

inequality solving
Solving inequalities is a fundamental aspect of algebra. It involves finding the values of a variable that make an inequality true. Here, we have an inequality \(x^3 > 4x^2\). The solution process involves several steps:

  • First, express the inequality clearly.
  • Simplify the inequality by performing algebraic operations.
  • Solve for the variable.
The main goal is to isolate the variable on one side.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In our example, we are working with \(x^3\) and \(4x^2\).

When dealing with algebraic expressions within inequalities, remember to:

  • Combine like terms if possible.
  • Use arithmetic operations to simplify the expression.
  • Ensure the variable is isolated for solving.


Simplifying algebraic expressions can often make complex inequalities much easier to solve.
polynomial inequalities
Polynomial inequalities involve polynomials, which are expressions consisting of variables raised to whole number powers. Inequalities like \(x^3 > 4x^2\) fall into this category.

To solve polynomial inequalities:

  • First, bring all terms to one side to form a single polynomial expression (e.g., \(x^3 - 4x^2 > 0\)).
  • Factor the polynomial, if possible.
  • Determine the critical points (where the polynomial equals zero).
  • Test intervals around these critical points to determine the sign of the polynomial.


In this case, dividing through by \(x^2\) and simplifying makes the solution more straightforward.

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

Prove the Rational Zeros Theorem. [Hint: Let \(\frac{p}{q},\) where \(p\) and \(q\) have no common factors except 1 and \(-1,\) be a zero of the polynomial function $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} $$ whose coefficients are all integers. Show that $$ a_{n} p^{n}+a_{n-1} p^{n-1} q+\cdots+a_{1} p q^{n-1}+a_{0} q^{n}=0 $$ Now, show that \(p\) must be a factor of \(a_{0}\), and that \(q\) must be a factor of \(\left.a_{n} .\right]\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the absolute maximum of \(f(x)=-\frac{2}{3} x^{2}+6 x-5\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Express the inequality \(3 \leq x<8\) using interval notation.

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=3 x^{4}+4 x^{3}+7 x^{2}+8 x+2 $$

Make up an inequality that has no solution. Make up one that has exactly one solution.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.