Question

$$x>2$$ $$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ x^{3} & 4 x \end{array}$$ A. Quantity A is greater. B. Quantity B is greater. C. the two quantities are equal. D. The relationship cannot be determined from the information given.

Short answer

A. Quantity A is greater.

Step-by-step solution

Understand the inequality

The given inequality is \( x > 2 \). This means that \( x \) is any real number greater than 2.

Express both quantities

The quantities given are Quantity A: \( x^3 \) and Quantity B: \( 4x \).

Analyze the behavior for \( x > 2 \)

We need to compare \( x^3 \) and \( 4x \). For \( x > 2 \), calculate \( x^3 \) and \( 4x \) for a few values of \( x \) greater than 2 to establish a pattern. For example: \( x = 3 \), \( x^3 = 27 \) and \( 4x = 12 \). For \( x = 4 \), \( x^3 = 64 \) and \( 4x = 16 \).

Compare the quantities

From the calculations, it is evident that as \( x \) increases, \( x^3 \) grows faster than \( 4x \). Hence, for \( x > 2 \), \( x^3 \) will always be greater than \( 4x \).

Conclusion

Therefore, Quantity A (\( x^3 \)) is greater than Quantity B (\( 4x \)).

Key concepts

Inequalities

Understanding inequalities is crucial in comparing two mathematical expressions. An inequality tells us that one value is larger or smaller than another.
This concept is essential in GRE quantitative comparison problems.
The given inequality states that \( x > 2 \).
This means that any value of \( x \) is more significant than 2. It encompasses all real numbers greater than 2, like 3, 4, or even 2.1.
Knowing this, we can compare expressions involving \( x \) with appropriate substitutions and understand the resultant behavior of these expressions.

Exponential Functions

Exponential functions involve expressions where the variable has an exponent, such as \( x^3 \).
When an increasing variable is raised to a power, its growth is rapid.
For example:

• When \( x = 3 \), \( x^3 = 27 \).
• When \( x = 4 \), \( x^3 = 64 \).

This rapid growth contrasts with linear functions like \( 4x \), where you multiply the variable by a constant factor.
Understanding this exponential growth helps in comparing expressions since exponential terms will eventually surpass linear terms for sufficiently large values of the variable.

Algebraic Expressions

An algebraic expression consists of variables, coefficients, and arithmetic operations. In this problem, we compare two expressions: \( x^3 \) and \( 4x \).

• \( x^3 \) is an exponential expression, where \( x \) is raised to the power of 3.
• \( 4x \) is a linear expression, where \( x \) is multiplied by 4.

To compare them effectively, substitute different values of \( x \) greater than 2 to determine their relative sizes.
This approach shows that as \( x \) increases beyond 2, the value of \( x^3 \) grows much faster than \( 4x \), indicating that \( x^3 \) will always be greater than \( 4x \) for any \( x > 2 \).