Question

Column A: \(\frac{2 + 3x}{2}\)
Column B: \(1 + 3x\)

• 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

Hint:

When comparing algebraic expressions, simplify them first. If the comparison depends on a variable, test three cases: a positive number, a negative number, and zero. If you get different results, the answer is (D).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is an algebraic comparison problem. The relationship between the two quantities depends on the value of the variable \(x\), which is not specified. We should simplify the expressions and then test different values for \(x\).
Step 2: Detailed Explanation:
Let's simplify the expression in Column A.
\[ \text{Column A} = \frac{2 + 3x}{2} = \frac{2}{2} + \frac{3x}{2} = 1 + 1.5x \] Now we compare the simplified Column A with Column B.
Column A: \(1 + 1.5x\)
Column B: \(1 + 3x\)
We can subtract 1 from both quantities without changing the comparison.
Compare: \(1.5x\) vs. \(3x\).
Now, the relationship depends entirely on the value of \(x\).
Step 3: Test different values for x:
Case 1: \(x\) is positive.
Let \(x = 2\).
Column A becomes \(1.5(2) = 3\).
Column B becomes \(3(2) = 6\).
In this case, Column B \textgreater Column A.
Case 2: \(x\) is negative.
Let \(x = -2\).
Column A becomes \(1.5(-2) = -3\).
Column B becomes \(3(-2) = -6\).
Since \(-3 \textgreater -6\), in this case, Column A \textgreater Column B.
Case 3: \(x\) is zero.
Let \(x = 0\).
Column A becomes \(1.5(0) = 0\).
Column B becomes \(3(0) = 0\).
In this case, Column A = Column B.
Step 4: Final Answer:
Since the relationship between the two quantities changes depending on the value of \(x\), the relationship cannot be determined from the information given.