Test: Inequalities - GMAT MCQ

Statement (1): ws + pt > 0 Statement (1) provides a relationship between ws, pt, and zero, but it does not provide any direct information about wp or st. Therefore, we cannot determine if wp + st > 0 based on this statement alone.

Statement (2): wt + ps > 0 Statement (2) provides a relationship between wt, ps, and zero, but it also does not provide any direct information about wp or st. Similar to Statement (1), we cannot determine if wp + st > 0 based on this statement alone.

When we consider both statements together, we still do not have any direct information about wp or st. Both statements provide relationships between different variables, but they do not provide any information that directly relates to wp or st. Therefore, even when combined, Statements (1) and (2) do not provide sufficient information to determine if wp + st > 0.

As a result, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1) says that n - 1 is divisible by 4, where m is an integer. However, this does not provide any specific information about the value of n. For example, if n = 18, then n - 1 = 17, which is divisible by 4. On the other hand, if n = 21, then n - 1 = 20, which is also divisible by 4. Therefore, statement (1) alone does not provide a unique value for n.

Statement (2) says that n² is less than 300. If n is a prime number, it cannot be equal to 17 since 17² = 289, which is not less than 300. However, statement (2) alone does not provide a definitive answer either since there are other prime numbers that satisfy the given condition, such as 2, 3, 5, 7, 11, 13, 19, 23, 29, etc.

Considering both statements together, we still cannot determine the exact value of n. Even though statement (1) suggests that n - 1 is divisible by 4, it does not guarantee that n is equal to 17. Additionally, statement (2) only provides an upper limit for n², but it does not exclude other prime numbers apart from 17.

Therefore, the correct answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

From statement 1:

3X < 2Y
2*3 and 3*2 are same.
Now, the decimal value is only dependent on whether 3X > 2Y.
As the statement says that 3X < 2Y
We can say that 3(2.00X) is always lesser than 2(3.00Y)
Sufficient.

From statement 2:

x+3 < y
So, Y is > X.
Then 3(2.00X) is always lesser than 2(3.00Y)
Sufficient.

D is the answer.

If g is prime then it will only have factors g and 1. If g is not prime then it will have factors besides g and 1.

Statement 1 is insufficient as there are multiple answers that could be prime/ non-prime.

Statement 2 is sufficient because
11!+11 and all the integers between up to 11!+2 all contain 11! which has all the factors of 11! i.e. 11x10x9x8...x3x2x1 such that adding any factor in range of 11! will only make the final number contain more factors of any given added number.

This plays out as follows:
If g=11!+11 then we can factor out 11, so g would be multiple of 11, thus not a prime;
If g=11!+10 then we can factor out 10, so g would be multiple of 10, thus not a prime;
If g=11!+9 then we can factor out 10, so g would be multiple of 9, thus not a prime;
If g=11!+8 then we can factor out 10, so g would be multiple of 8, thus not a prime;
If g=11!+7 then we can factor out 10, so g would be multiple of 7, thus not a prime;
If g=11!+6 then we can factor out 10, so g would be multiple of 6, thus not a prime;
If g=11!+5 then we can factor out 10, so g would be multiple of 5, thus not a prime;
If g=11!+4 then we can factor out 10, so g would be multiple of 4, thus not a prime;
If g=11!+3 then we can factor out 10, so g would be multiple of 3, thus not a prime;
If g=11!+2 then we can factor out 10, so g would be multiple of 2, thus not a prime;

Thus all numbers we can factor out contain a factor other than g and 1; thus g is not prime.

Thus, statement 2 tells us g has more than g and 1 as its factors. Thus statement 2 is sufficient.

Statement (1) states that x is greater than x³. For this statement to hold true, x³ must be negative. Since x is an integer, the only way for x³ to be negative is if x is negative. If x is negative, it cannot be positive. Therefore, statement (1) alone is sufficient to determine that x is not positive.

Statement (2) states that x is less than x². This inequality does not provide enough information to determine if x is positive or negative. For example, if x = -2, then x² = 4 and x is negative. However, if x = 1/2, then x² = 1/4 and x is positive. Statement (2) alone is not sufficient to answer the question.

By considering statement (1) alone, we can conclude that x is not positive. Therefore, statement (1) alone is sufficient to answer the question.

Hence, the correct answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1) states that x is less than 0. Given that y = |x| + x, we can see that |x| is always non-negative. Since x is negative, |x| will be equal to the positive value of x, resulting in |x| + x being equal to 0. Therefore, if x is less than 0, y will be equal to 0. Statement (1) alone is sufficient to answer the question.

Statement (2) states that y is less than 1. However, this does not provide enough information to determine if y is equal to 0. It is possible for y to be less than 1 but not equal to 0. For example, if x = -0.5, then y = |-0.5| + (-0.5) = 0.5 - 0.5 = 0, which satisfies y = 0. However, if x = -1, then y = |-1| + (-1) = 1 - 1 = 0, which also satisfies y = 0. Hence, statement (2) alone is not sufficient to answer the question.

By considering both statements separately, we see that statement (1) alone is sufficient to determine that y is equal to 0. Therefore, each statement alone is sufficient to answer the question.

Thus, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1) states that x² is greater than 9. If x² > 9, it means that x² is positive (since the square of any real number is non-negative). This tells us that x cannot be zero or positive. Since we are specifically looking for negative values of x, we can infer that x must be less than -3. Statement (1) alone is sufficient to answer the question.

Statement (2) states that x² is less than -9. This does not provide enough information to determine if x is less than -3. For example, if x = -2, then x³ = -8, which is greater than -9. However, if x = -4, then x³ = -64, which is also less than -9. Statement (2) alone is not sufficient to answer the question.

By considering statement (1) alone, we can conclude that x is less than -3. Therefore, statement (1) alone is sufficient to answer the question.

Hence, the correct answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1) does not provide enough information to determine the value of the largest number. For example, the three prime numbers could be 3, 5, and 7, in which case the largest number is 7. However, they could also be 7, 11, and 13, in which case the largest number is 13.

Statement (2) is more helpful. If the product of the three numbers is less than 1310, then the largest number must be less than the cube root of 1310, which is approximately 11. Therefore, the largest number must be one of the prime numbers less than or equal to 11.

Combining the two statements, we know that the three numbers are prime and that the largest number is less than or equal to 11. Therefore, the largest number must be either 5, 7, or 11. However, we still cannot determine which of these is the largest number.

Therefore, the answer is (E) the information provided is not sufficient to determine the value of the largest number.

Statement (1) states that X - Z + Y > 0. This can be rearranged as X > Z - Y. Since Z and Y are positive integers, Z - Y will be negative or zero. Therefore, X will be greater than Z - Y. Statement (1) alone is sufficient to answer the question.

Statement (2) states that Z² = X² + Y². This is a Pythagorean triple equation where Z is the hypotenuse and X and Y are the other two sides. While this equation provides a relationship between X, Y, and Z, it does not directly tell us if X is greater than Z - Y. Statement (2) alone is not sufficient to answer the question.

By considering both statements together, we know that X > Z - Y from statement (1), and we have the Pythagorean relationship from statement (2). From the Pythagorean relationship, we can determine the relative values of X, Y, and Z but not specifically if X is greater than Z - Y. Therefore, both statements together do not provide a definitive answer.

Thus, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.