Question

Given \(x>0\).
Quantity A: \(-5x + 4\)
Quantity B: \(8 - 2x\)

• A. The relationship cannot be determined from the information given.
• B. The two quantities are equal.
• C. Quantity A is greater.
• D. Quantity B is greater.
• E.

Hint:

For quantitative comparisons, solving the inequality algebraically is the most rigorous method. However, testing one or two simple values that fit the given conditions can give you a strong indication of the correct answer and help you catch potential errors in your algebra.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a quantitative comparison question. We need to compare two algebraic expressions given a constraint on the variable x. We can do this by setting up an inequality and solving for x, or by analyzing the relationship between the two expressions.
Step 2: Key Formula or Approach:
To compare Quantity A and Quantity B, we can subtract one from the other or set up an inequality. Let's find out for which values of x Quantity B is greater than Quantity A.
\[ \text{Quantity B}>\text{Quantity A} \] \[ 8 - 2x>-5x + 4 \] Step 3: Detailed Explanation:
Let's solve the inequality: \[ 8 - 2x>-5x + 4 \] Add \(5x\) to both sides: \[ 8 + 3x>4 \] Subtract 8 from both sides: \[ 3x>4 - 8 \] \[ 3x>-4 \] Divide by 3: \[ x>-\frac{4}{3} \] The inequality \(8 - 2x>-5x + 4\) is true whenever \(x>-4/3\).
The problem gives us the condition that \(x>0\).
Since any number greater than 0 is also greater than -4/3, the condition \(x>-4/3\) is always satisfied for any given \(x>0\).
Therefore, for all \(x>0\), Quantity B is always greater than Quantity A.
Alternative Method (Testing Values):
Let's pick a value for x that satisfies \(x>0\), for example, \(x = 1\).
Quantity A = \(-5(1) + 4 = -1\)
Quantity B = \(8 - 2(1) = 6\)
In this case, Quantity B>Quantity A.
Let's pick another value, \(x = 10\).
Quantity A = \(-5(10) + 4 = -46\)
Quantity B = \(8 - 2(10) = -12\)
In this case, Quantity B>Quantity A as well. This suggests that Quantity B is always greater. The algebraic method confirms this.
Step 4: Final Answer:
Quantity B is greater.