Question

Is xy $>$ 24?
1. y - 2 $<$ x
2. 2y $>$ x + 8

• A. EACH statement ALONE is sufficient to answer the question asked
• B. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• C. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• D. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• E. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

When combining two inequalities in a Data Sufficiency problem, first try to establish a relationship between the variables, as in $\frac{x}{2} + 4 < y < x + 2$. Then, check the condition for this relationship to be possible (here, $x>4$). Finally, use this new constraint to evaluate the expression in the question.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a data sufficiency question involving inequalities. We need to determine if the product \(xy\) is definitively greater than 24. We must test if the given statements constrain \(x\) and \(y\) in such a way that \(xy > 24\) is always true or always false.

Step 2: Key Formula or Approach:
We will analyze the inequalities provided in each statement. A statement is sufficient if it leads to a definite "Yes" or a definite "No" answer. If we can find examples that give both "Yes" and "No" answers, the statement is not sufficient.

Step 3: Detailed Explanation:
Analyzing Statement (1): y - 2 $<$ x
This inequality can be rewritten as \(y < x + 2\).
Let's test some cases: - Case 1 (Answer "Yes"): If \(x=10\), then \(y < 12\). Let's choose \(y=10\). Then \(xy = 100\), and \(100 > 24\). - Case 2 (Answer "No"): If \(x=4\), then \(y < 6\). Let's choose \(y=5\). Then \(xy = 20\), and \(20\) is not greater than 24.
Since we can get both "Yes" and "No" answers, Statement (1) ALONE is not sufficient.
Analyzing Statement (2): 2y $>$ x + 8
This inequality can be rewritten as \(y > \frac{x}{2} + 4\).
Let's test some cases: - Case 1 (Answer "Yes"): If \(x=10\), then \(y > \frac{10}{2} + 4\), so \(y > 9\). Let's choose \(y=10\). Then \(xy = 100\), and \(100 > 24\). - Case 2 (Answer "No"): If \(x=2\), then \(y > \frac{2}{2} + 4\), so \(y > 5\). Let's choose \(y=6\). Then \(xy = 12\), and \(12\) is not greater than 24.
Since we can get both "Yes" and "No" answers, Statement (2) ALONE is not sufficient.
Analyzing Statements (1) and (2) Together:
We have a system of two inequalities: 1. \(y < x + 2\) 2. \(y > \frac{x}{2} + 4\) Combining them, we get: \[ \frac{x}{2} + 4 < y < x + 2 \] For this range of \(y\) to exist, the lower bound must be less than the upper bound: \[ \frac{x}{2} + 4 < x + 2 \] Subtract \(\frac{x}{2}\) from both sides: \[ 4 < \frac{x}{2} + 2 \] Subtract 2 from both sides: \[ 2 < \frac{x}{2} \] Multiply by 2: \[ 4 < x \] So, a necessary condition for both statements to hold is that \(x\) must be greater than 4. (Assuming x and y are positive, which is standard unless specified).
Now, let's look at the product \(xy\). We can use the inequality for \(y\): \[ xy > x \left(\frac{x}{2} + 4\right) \] \[ xy > \frac{x^2}{2} + 4x \] Let's analyze the expression \(\frac{x^2}{2} + 4x\). We know that \(x > 4\). Let's see what the minimum value of this expression is for \(x > 4\). Let \(f(x) = \frac{x^2}{2} + 4x\). Since this is an upward-opening parabola, its value increases for \(x > -4\). As our domain is \(x > 4\), the expression will be minimized as \(x\) approaches 4 from the right. Let's evaluate the expression at \(x=4\): \[ f(4) = \frac{4^2}{2} + 4(4) = \frac{16}{2} + 16 = 8 + 16 = 24 \] Since we know that \(x > 4\), it must be true that \(\frac{x^2}{2} + 4x > 24\). Therefore, we have: \[ xy > \frac{x^2}{2} + 4x > 24 \] This proves that \(xy\) must always be greater than 24. The answer to the question is a definite "Yes".
Therefore, both statements TOGETHER are sufficient.

Step 4: Final Answer:
Neither statement is sufficient on its own, but when combined, they provide enough information to definitively answer the question. The correct option is (B).