Neither statement alone is sufficient, as in each case the x and y terms can appear in any part of the number line (both large positive numbers, both large negative numbers - each of which would give a "yes" answer - or one of each, in which case you'd have a negative product and a "no answer). So the problem really "gets started" when you take the statements together to assess "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient" vs. "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

In doing so, the common mistake that most students make is to pick numbers to reason out a solution rather than apply conceptual understanding and algebraic manipulation. Number picking has its place on some problems (some can only be solved using numbers to show patterns, etc.) but most can be solved quickly and efficiently with algebra. Using your understanding of combining inequalities, it is possible to isolate x and y and learn more about them individually. First let’s eliminate x and isolate y:

Step 1: Rewrite the inequalities to line up variables

y – x < 2

2y – x > 8

Step 2: Multiply top inequality by -1 to get the signs pointing the same way and then combine to eliminate x:

-y + x > -2

2y – x > 8

y > 6

Repeat step 2 to eliminate y by multiplying the top inequality by -2 to get the signs pointing the same way and then combine:

-2y + 2x > -4

2y - x > 8

x > 4

If y > 6 and x > 4 then you know that the product of xy must be greater than 24 and the answer to the question is "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient".

This problem features a classic example of "algebraic mirroring," a technique via which you use algebraic manipulation to either make one of the statements look like the question or make the question look like one of the statements.

With statement 1, your clue that you should use algebraic mirroring is that number 16 that appears as a coefficient in statement 1 and as a constant in the question. Can you isolate 16 on the right hand side of the equation in statement 1 to make it look like the question? If you divide both sides by  you can:

becomes

And from here you can use a bit of "reverse engineering" with the process of adding fractions. If you were to add two fractions, you'd find a common denominator and then sum the numerators; by that same logic, since you have a "common denominator" (one single denominator) of , you can break apart the numerator:

This means that you can rephrase the equation as:

And then you can factor out the common terms in each fraction:

This directly mirrors the question, so you know that the answer is "yes" and statement 1 is sufficient.

Alternatively, you could manipulate the question to look like the statement. If you take

 and multiply each side by  the question then asks:

Is 

And the fractions factor to:

Is ?

To which statement 1 loudly proclaims "yes." Again, statement 1 is sufficient.

Statement 2 is not sufficient, as it allows for both "yes" and "no" answers. When you equate  and  to make the question look like:

The answer is "yes" if  and "no" if the value is anything else, so statement 2 is not sufficient and the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".  

This exponent-based problem involves an important lesson in Data Sufficiency strategy: if the problem asks for the value formed by a combination of variables (such as here), there is usually a way to solve specifically for that combination without having to solve for the variables individually!

Here you can do that given statement 1. If you employ the first guiding principle of exponents, "find common bases," you can factor the 3, 9, and 27 all into base 3s so that all your bases are common. That means that:

Which then means that you can employ the rule for taking one exponent to another (in other words, ) and rephrase this as:

Then you can combine the terms on the left using the rule that when you multiply two exponents with the same base, you add the exponents. Therefore:

And here you can set the exponents equal using another law of exponents, meaning that , and making Statement 1 sufficient.

Statement 2 is not sufficient, as merely knowing that  alone does not allow you to find a specific value for . Therefore the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

This yes/no data sufficiency question asks is 

Statement 1 states that . 

This statement may not appear sufficient at first, since it cannot be easily solved to give a value for  or for . However, it's always important to remember that sometimes things that would be insufficient to solve a "what is the value" question may be perfectly suitable for a "yes/no" question.

In this case, look to simplify the question stem using what you know from statement (1). Since statement (1) matches what's on the right hand side of the equation in the question stem, you can substitute it in to get:

Is 

A squared number cannot be negative, so the answer must always be yes. This statement is consistent so it is sufficient. Eliminate "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked", "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", and "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

Statement (2) gives that  and 

Again, this statement may not appear to be sufficient. It does not give specific values for  or . However, if you “Just Do It” or if you plug in the numbers then you will see that it is sufficient.

Conceptually it looks like this: so long as  is greater than 3,  will be greater than . Whatever number  is can only take away from the - it cannot add to it. In fact, statement (2) gives more information than strictly necessary;  would have been sufficient. The correct answer is "EACH statement ALONE is sufficient to answer the question asked".

If this wasn't apparent, you could also pick numbers. Remember as you plug in numbers, however, that you want to test the limits of the problem to try to force statement (2) to be inconsistent. One way to do this is to pick extreme numbers for  and . 

If  and , the statement becomes

Is  Since  is definitely bigger than , the answer is yes.

But what if  were bigger than 

If  and ,  the problem becomes

Is . Since  is obviously much bigger than any negative number, the answer is also yes. Statement (2) is sufficient. Eliminate "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" and choose answer choice "EACH statement ALONE is sufficient to answer the question asked".

There are essentially two ways to manipulate the algebra for statement 1 to get it to look a bit more like the question stem.

One way is to multiply top and bottom by 

 means that in the top set of fractions   yields simply . And for the bottom, the  terms will cancel, leaving just 

. So the new fraction is:  And here's where one more step working with fractions really pays off. You can express that as  which leaves you with .

This question asks for a specific number value for x given that x is a two-digit integer.

 Statement (1) states that the product of the two digits is 14. To figure out potential digits that would fit this statement, consider that factor-pairs of 14: 1 and 14, and 2 and 7. Notice that 1 and 14 isn't an option since you're looking for single-digit numbers to make up a two-digit number. Therefore, the two digits must be 2 and 7. One possibility for the value of x is therefore 27.

However, you must remember to Play Devil's Advocate. What happens if you switch the positions of the digits? 72 still fulfills the constraints of the statement and is also therefore a possible value of x. You must therefore conclude that Statement (1) is not sufficient and eliminate "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" and "EACH statement ALONE is sufficient to answer the question asked".

Statement (2) gives that x is divisible by 9. In other words, x is a multiple of 9. However, there are many two-digit multiples of 9, such as 18, 27, 36, etc. This is clearly not sufficient since the statement can yield multiple values for x. Eliminate choice "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

When you take the statements together, you should recognize that the two, separate values you found in statement (1) still work here. Both 27 and 72 are multiples of 9, and both have digits that multiply to 14. Because the two statements together still give you two separate values, you must conclude that the two statements are not sufficient. The correct answer is " Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

In beginning this problem, it is a good idea to take inventory of everything you know. For one, you can sum the "known" values: 3 + 8 + 17 = 28. Secondly, you can use the given equation  to put the entire list in terms of : if , then the sum of the values:

Can be expressed as 

And then you can also note that for the average of 5 terms to be less than 8, the sum of those terms must be less than 40. So this question is also asking:

Is ?

Which simplifies to:

Is 

Is 

As you progress to the statements, statement 1 may look tempting: If , wouldn't the least possible value of  be 3?

No - that's only true if you know that  is an integer, which is not necessarily the case.  could be something like 2.5, which would give the answer "yes" to the question. Or  could of course be anything else greater than 2 (like 10 or 100), which would give the answer "no." So statement 1 is not sufficient. And there is an important lesson here: you cannot assume that a value is an integer unless you're either told so or given information that proves it so.

Statement 2, however, is sufficient. When you factor down  that means that . This means that  must be less than 3, guaranteeing that the answer to the question is "yes." Therefore the correct answer is "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

Importantly, this is a Yes/No Data Sufficiency question. In such a question, even if there are multiple solutions for a variable you will have sufficient information if all solutions provide the same answer ("always Yes" or "always No"). For this reason, it is always important to note whether you're dealing with a "what is the value?" question or a "Yes/No" question.

Here you're asked whether  is greater than 

 With statement 1, you're provided with a quadratic. But even though quadratics generally yield multiple solutions, note that a quadratic as a statement can provide sufficient information if:

-All solutions provide the same answer (as you'll see here)

-It is a "special" quadratic that factors to only one solution (in the form  or  )

 For this reason, you should make a point of always doing (or at least beginning) the math on quadratics to see what the quadratic will yield. Here you can start by subtracting  from both sides to yield: 

And then factor to:

When you solve, by setting each parenthetical equal to zero, you'll have:

 or  

Note that each possible value for x is less than 9, meaning that each gives the answer "no." So while you don't know exactly what x is, you do have sufficient information to determine that the answer to the question is "no," so statement 1 is sufficient. Consequently you can eliminate choices "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked", "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", and Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed" 

With statement 2, savvy examinees will note that with a linear equation there can be only one solution. Here you can save time - if you know that the equation will yield exactly one value, then that exact value will guarantee exactly one answer to the overall question (that value will either be greater than 9, so "yes," or it will not, so "no"). Therefore this statement must be sufficient, yielding the answer "EACH statement ALONE is sufficient to answer the question asked"

Of course, if you do not see that at first, there is no harm in doing the math. If you subtract  from each side of   you will have: 

Then dividing both sides by 4 will yield , guaranteeing the answer "no."

 Note here, also, that your job is only to determine whether the information is sufficient to answer the question, not whether that answer happens to be "yes." So a consistent, guaranteed "no" answer means that the information is "sufficient."

This problem is a classic example of how the GMAT tests "precision in language." The question asks about the average sale in each month, but the data you're given is about the total revenue from sales. And of course total revenue comes from adding up all of the individual sales in a month. The average sale would be calculated as 

As you assess the statements, it should be clear that neither is sufficient alone; each tells you only about one of the two months, leaving the other month completely undefined.

And even when you take them together, note that you have all the total sales revenue information you need for the numerators of the average sale calculation, but none of the "number of sales" information necessary to assess the exact question being asked. Therefore you do not have sufficient information to answer the question, and the correct answer is "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed". The lesson? Pay close attention to the specific question being asked; as you can see from the statistics, those who miss this problem pick "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", almost always because they compare "Month M to Month J" but along the total revenue metric, when that's not what the question asks.

This problem rewards those who leverage all assets in a problem. Because of the yes/no format, you do not need to know the exact measure of angle a (or any angle in the triangle) as long as you can prove that a is either always greater than 85 ("yes") or always less than (or equal to) 85 ("no").

Statement 1 is not sufficient, as it allows for each possibility of yes and no. But statement 2 is sufficient. Note that all angles in a triangle must equal 180. So if c is greater than 95, then a + b must be less than 85 in order for the total sum to be 180. Since a and b cannot be negative, they each must be less than 85, meaning that the answer is a definite "no." Therefore statement 2 is sufficient and the correct answer is "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".