Question

If \(yz \neq 0\), is \(x-y+z < xz-yz-xy\)?
1. \(\displaystyle \frac{x}{y}<-\frac{1}{2}\)
2. \(\displaystyle xy< 0\)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not 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. EACH statement ALONE is sufficient to answer the question asked
• D. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Hint:

In GMAT inequality problems, especially those in Data Sufficiency, be very mindful of variables whose signs are not determined. Here, the variable 'z' could be positive or negative, which dramatically changes the inequality. Testing both a positive and a negative value for an unconstrained variable is a key strategy.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a data sufficiency question involving an inequality with three variables x, y, and z. We need to determine if the given statements provide enough information to definitively answer "yes" or "no" to the inequality \(x-y+z < xz-yz-xy\).

Step 2: Key Formula or Approach:
The best approach for complex inequalities in data sufficiency is often to test cases. We will choose numbers that satisfy the given conditions and see if we can get both "yes" and "no" answers to the main question. If we can, the information is not sufficient.
The inequality can be rearranged: Is \(x-y+z - (xz-yz-xy) < 0\)?
Is \(x-y+z - xz+yz+xy < 0\)?
Let's analyze the statements.
Statement 2: \(xy < 0\) means x and y have opposite signs.
Statement 1: \(\frac{x}{y} < -\frac{1}{2}\). This is a stronger condition than statement 2. If \(\frac{x}{y}\) is negative, then \(xy\) must also be negative. So, statement 1 implies statement 2.

Step 3: Detailed Explanation:
Analyzing Statement (2): \(xy< 0\).
This means x and y have opposite signs. We don't know anything about z, except that \(z \neq 0\).
Let's test some numbers.
Case 1: Let \(x=1, y=-1\). Then \(xy = -1 < 0\). Let \(z=1\).
Is \(1 - (-1) + 1 < 1(1) - (-1)(1) - 1(-1)\)?
Is \(3 < 1 + 1 + 1\)? Is \(3 < 3\)? This is false. So the answer is "No".
Case 2: Let \(x=2, y=-1\). Then \(xy = -2 < 0\). Let \(z=1\).
Is \(2 - (-1) + 1 < 2(1) - (-1)(1) - 2(-1)\)?
Is \(4 < 2 + 1 + 2\)? Is \(4 < 5\)? This is true. So the answer is "Yes".
Since we can get both "Yes" and "No" answers, statement (2) alone is not sufficient.
Analyzing Statement (1): \(\displaystyle \frac{x}{y}<-\frac{1}{2}\).
This condition implies \(xy < 0\). Let's test cases that satisfy this stricter condition. The variable \(z\) is still unconstrained (other than \(z \neq 0\)). The sign of z can significantly impact the inequality.
Let's use a case that satisfies statement (1) and see how changing \(z\) affects the result.
Let \(x=2, y=-1\). Then \(\frac{x}{y} = \frac{2}{-1} = -2\), which is less than \(-\frac{1}{2}\). This case satisfies the condition.
Case 1.1: Let \(z = 1\).
Is \(2 - (-1) + 1 < 2(1) - (-1)(1) - 2(-1)\)?
Is \(4 < 2 + 1 + 2\)? Is \(4 < 5\)? Yes.
Case 1.2: Let \(z = -1\).
Is \(2 - (-1) + (-1) < 2(-1) - (-1)(-1) - 2(-1)\)?
Is \(2 < -2 - 1 + 2\)? Is \(2 < -1\)? No.
Since we can get both "Yes" and "No" answers with statement (1), it alone is not sufficient.
Analyzing Both Statements Together:
As noted earlier, statement (1) implies statement (2). Therefore, combining the statements provides no new information beyond what statement (1) already gives us. Since statement (1) alone was not sufficient, both statements together are also not sufficient. Our test with \(x=2, y=-1\) satisfied both statements, and we found that the answer to the question depended on the value of \(z\).

Step 4: Final Answer:
Statements (1) and (2) together are not sufficient to answer the question asked.