Question

Is 2x + 1 $>$ 0.
1. x is an integer
2. $|x| < 1.5$

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is 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:

For Yes/No questions, test boundary cases and representative numbers within the given constraints. If you find one case that gives a "Yes" and another that gives a "No", the information is insufficient.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a "Yes/No" Data Sufficiency question involving an inequality. We need to determine if the given conditions are sufficient to conclude definitively whether \( 2x + 1 > 0 \).
First, let's simplify the inequality:
\( 2x + 1 > 0 \)
\( 2x > -1 \)
\( x > -0.5 \)
So the question is: "Is x greater than -0.5?"

Step 2: Detailed Explanation:
Analyze Statement (1): "x is an integer."
Integers can be greater or less than -0.5.
- If we choose \( x = 1 \) (an integer), then \( 1 > -0.5 \). The answer is "Yes".
- If we choose \( x = -1 \) (an integer), then \( -1 \) is not greater than \( -0.5 \). The answer is "No".
Since we can get both "Yes" and "No" answers, Statement (1) is not sufficient.
Analyze Statement (2): "$|x| < 1.5$."
This inequality means that x is between -1.5 and 1.5.
\[ -1.5 < x < 1.5 \] This range contains values that are both greater than and less than -0.5.
- If we choose \( x = 1 \), then \( -1.5 < 1 < 1.5 \) is true, and \( 1 > -0.5 \). The answer is "Yes".
- If we choose \( x = -1 \), then \( -1.5 < -1 < 1.5 \) is true, but \( -1 \) is not greater than \( -0.5 \). The answer is "No".
Since we can get both "Yes" and "No" answers, Statement (2) is not sufficient.
Analyze Statements (1) and (2) Together:
We need to find values of x that satisfy both conditions:
1. x is an integer.
2. \( -1.5 < x < 1.5 \).
The integers that fall within this range are -1, 0, and 1.
Let's test these values for the original question "Is \( x > -0.5 \)?":
- If \( x = -1 \), is \( -1 > -0.5 \)? No.
- If \( x = 0 \), is \( 0 > -0.5 \)? Yes.
- If \( x = 1 \), is \( 1 > -0.5 \)? Yes.
Even with both statements combined, we can still get a "No" answer (when x=-1) and a "Yes" answer (when x=0 or x=1). Because we cannot arrive at a single, definitive conclusion, the information is not sufficient.

Step 3: Final Answer:
Since even together the statements do not provide a definitive answer, the correct option is (E).