Is k² + k – 2 > 0?
(1) k is an integer greater than zero.
(2) k divided by 2 is an integer.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) The first thing we need to do is factor k² + k – 2 into (k + 2)(k – 1). From these factors, we can see that the value of k² + k – 2 = 0 when k = -2 or 1.

When k > 1, both k + 2 and k – 1 will be positive, so the product of the two (the original expression) will be positive.

Similarly, when k is between -2 and 1, the product of the two expressions will be negative.

Finally, when k is less than -2, the product will be positive.

So we've learned that k² + k – 2 > 0 as long as k is greater than 1 or less than -2.

Statement (1) tells us that k is an integer greater than zero. This is not sufficient because k² + k – 2 > 0 when k is greater than 1, but it will equal zero when k is 1.

Statement (2) tells us k is an even integer. This is also insufficient, because all even integer values of k will make k² + k – 2 greater than zero except -2 and 0. k = -2 will make the expression equal to zero, and k = 0 will make the expression negative. (Remember: 0 is an even integer!)

Combined, the two statements are sufficient, because the only possible values for k are positive even numbers (2 or greater), all of which make k² + k – 2 > 0.

Since the statements are insufficient individually, but sufficient when combined, the correct answer is choice (C).
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This question doesnt make any sense with your explanation.

What exactly don't you understand?

Here are the key moments for your consideration:
1. We solve the inequality. The solution can be displayed as:


2. We try statement (1). It is NOT sufficient, because k² + k – 2 can be positive, but it can be zero as well (if k = 1). We have two different answers to the main question, so it is NOT sufficient.

3. We try statement (2). It is NOT sufficient, because k² + k – 2 can be positive, negative or 0.

4. The both statements combined define k to be a positive even integer (2, 4, 6 ....). All this values make k² + k – 2 to be positive. So the answer to our main question is always "YES". Therefore the both statements combined are sufficient to answer the question.

We could use the number substitution method to solve this problem.
The question says is k² + k – 2 > 0

1. k > 0
since 1 is the smallest integer substitute the value of k = 1 in the equation, we'll get:
(1)² + 1 – 2 > 0
= 1 + 1 – 2 > 0
= 2 – 2 > 0
= 0 > 0

Since the above statement is incorrect, 1 is NOT SUFFICIENT.

2. k is an even integer.
The condition states that k is even. So substitute the value of k with +2 and -2.
For +2 we get,
(2)² + 2 – 2 > 0
= 4 > 0
For -2 we get,
(-2)² + (-2) – 2 > 0
= 4 – 4 > 0
= 0 > 0

Since the above statement is incorrect, 2 is NOT SUFFICIENT.

Considering both 1&2, we get
k is a positive even integer.
By substituting the value of k as +2 as shown above we get,
4 > 0 .
To cross check this answer substitute the value of k as 4 in the the equation. We'll get
(4)² + 4 – 2 > 0
= 16 + 2 > 0
= 18 > 0

Since the above condition hold to be true for all values of k > 2, both 1&2 together ARE SUFFICIENT to solve the question. Therefore C is the correct option.

Note, that for the substitution method to work in a "Yes/No" data sufficiency question we need to achieve the following:

1) In order to prove that information is not sufficient, we need to plug in a value that yields "YES" as an answer, and a value that yields "NO" as an answer.

2) In order to prove that information is sufficient you must plug in ALL the possible values and make sure that they all yield the same answer (always "YES" or always "NO"). Thus this method is seldom used to prove sufficiency.

You have shown that one possible value yields "NO" as the answer. But you know nothing about other possible values. If all the possible values yield "NO" as the answer, then this information will be sufficient.

Statement (2) is incorrect, because you've shown that one value yields "YES" and the other one yields "NO".

Substituting just two out of infinitely many possible values can not be a prove of sufficiency.
You have not shown that the answer is "YES" for ALL the possible values of k (positive even integers).

Try to feel the logic of a "YES/NO" data sufficiency question and how the number substitution method should be used here (to prove insufficiency mostly).