Question

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

Short answer

The positive numbers greater than 4.

Step-by-step solution

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: \[x^3 > 4x^2\]

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: \[x > 4\]

Identify the Solution Set

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

Key concepts

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.