Question

w > 1
Quantity A: \( 7w - 4 \)
Quantity B: \( 2w + 5 \)

• 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.
• E.

Hint:

For quantitative comparisons, if you simplify the comparison and find it depends on how the variable relates to a specific number (in this case, 1.8), check if the given condition (w \textgreater 1) is sufficient to fix that relationship. If not, the answer is likely (D). Testing values on either side of that critical number is a great way to confirm this.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to compare two linear expressions of a variable 'w', given the condition that w>1. We can either simplify the comparison algebraically or test values that satisfy the condition.
Step 2: Key Approach (Algebraic Simplification):
Let's set up a comparison between the two quantities and simplify it to isolate 'w'.
\[ 7w - 4 \quad \Box \quad 2w + 5 \] Subtract \(2w\) from both sides:
\[ 5w - 4 \quad \Box \quad 5 \] Add 4 to both sides:
\[ 5w \quad \Box \quad 9 \] Divide both sides by 5 (a positive number, so the relationship does not change):
\[ w \quad \Box \quad \frac{9}{5} \] Since \( \frac{9}{5} = 1.8 \), the original comparison is equivalent to comparing w with 1.8.
Step 3: Detailed Explanation (Testing Values):
The simplified comparison depends on whether w is greater than, less than, or equal to 1.8. The only information we are given is that \(w>1\). This condition allows for w to be on either side of 1.8. Let's test two cases.
Case 1: Choose a value for w such that \(1<w<1.8\). Let's pick \(w = 1.5\).
- Quantity A = \( 7(1.5) - 4 = 10.5 - 4 = 6.5 \).
- Quantity B = \( 2(1.5) + 5 = 3 + 5 = 8 \).
In this case, Quantity B (8) is greater than Quantity A (6.5).
Case 2: Choose a value for w such that \(w>1.8\). Let's pick \(w = 2\).
- Quantity A = \( 7(2) - 4 = 14 - 4 = 10 \).
- Quantity B = \( 2(2) + 5 = 4 + 5 = 9 \).
In this case, Quantity A (10) is greater than Quantity B (9).
Step 4: Final Answer:
Since we found one case where Quantity B is greater and another case where Quantity A is greater, the relationship between the two quantities is not constant and depends on the specific value of w. Therefore, the relationship cannot be determined from the information given. The correct answer is (D).