Understanding Data Sufficiency and Inequality Questions

Data Sufficiency questions test your ability to determine if the information given in the statements is enough to answer the question asked. You don't need to find the exact value of a variable, just whether the statements provide sufficient information to answer the question definitively (either always YES or always NO).

The question asks: Is \(Y > 0\)?

We are given two statements:

• Statement 1: \(X + Y > 0\)
• Statement 2: \(X - Y > 0\)

Let's evaluate each statement separately, and then combine them, to see if we can definitively answer whether \(Y > 0\).

Analysing Statement 1 for Sufficiency

Statement 1 says \(X + Y > 0\). This means the sum of X and Y is a positive number.

Let's test some values for X and Y that satisfy this condition and see if Y must be greater than 0.

In the first example, \(Y\) is positive. In the second example, \(Y\) is negative. Since we can find cases where \(Y > 0\) is true and cases where \(Y > 0\) is false, Statement 1 alone is not sufficient to answer the question.

Checking Statement 2 Alone

Statement 2 says \(X - Y > 0\). This means X is greater than Y, or \(X > Y\).

Let's test some values for X and Y that satisfy this condition and see if Y must be greater than 0.

In the first example, \(Y\) is positive. In the second example, \(Y\) is negative. Since we can find cases where \(Y > 0\) is true and cases where \(Y > 0\) is false, Statement 2 alone is not sufficient to answer the question.

Combining Statements 1 and 2 for Sufficiency

Now, let's assume both statements are true:

1. \(X + Y > 0\)
2. \(X - Y > 0\)

We can add these two inequalities together:

\((X + Y) + (X - Y) > 0 + 0\)

\(2X > 0\)

Dividing by 2 (a positive number), we get \(X > 0\). So, if both statements are true, X must be positive.

From Statement 2, we also know \(X > Y\).

So, we know \(X\) is positive and \(X\) is greater than \(Y\). Does this guarantee that \(Y\) is positive?

Let's test some values that satisfy both \(X > 0\) and \(X > Y\), and also satisfy the original statements:

We found examples where both statements are true, and \(Y\) can be positive (\(Y=1\)), zero (\(Y=0\)), or negative (\(Y=-1\)). Since \(Y\) can be positive or not positive while satisfying both statements, the statements together are not sufficient to answer the question definitively.

Conclusion on Data Sufficiency

Based on our analysis:

• Statement 1 alone is not sufficient.
• Statement 2 alone is not sufficient.
• Statements 1 and 2 together are not sufficient.

Therefore, neither statement alone nor both statements together are sufficient to answer the question "Is \(Y > 0\)?".

Revision Table: Evaluating Sufficiency

Additional Information on Inequalities

Inequalities are mathematical statements that compare two expressions using symbols like \( > \) (greater than), \( < \) (less than), \( \ge \) (greater than or equal to), or \( \le \) (less than or equal to).

When working with inequalities:

• You can add or subtract the same number or expression from both sides without changing the direction of the inequality. For example, if \(a > b\), then \(a+c > b+c\).
• You can multiply or divide both sides by the same positive number without changing the direction of the inequality. For example, if \(a > b\) and \(c > 0\), then \(ac > bc\).
• If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality. For example, if \(a > b\) and \(c < 0\), then \(ac < bc\).
• Adding two inequalities: If \(a > b\) and \(c > d\), then \(a+c > b+d\). This is what we did when combining the two statements in the problem.

Data Sufficiency problems often require you to explore different cases or use properties of numbers and inequalities to determine if a definitive answer is possible.