Test: Inequalities - GMAT MCQ

Statement 1: “-5x – 3 > -2x” can be simplified by adding 3 to both sides to yield “ -5x > -2x + 3” and then adding 2x to both sides to yield “-3x > 3.” Then, to isolate x, divide both sides by 3 and –x > 1. To change the sign of x, multiply both sides by a negative 1 and the statement becomes “x < -1.” This algebraic manipulation has answered the question directly. “No, x is a not positive. x is a negative number.” This statement is sufficient. The answer is either A or D.

Statement 2: x² is positive allows for x to be either positive or negative. Since the answer is not consistent this statement is not sufficient.

The correct answer is A.

Statement (1): b ≥ 4

This statement tells us that b is greater than or equal to 4. If b is greater than or equal to 4, then the fraction 4/b would be less than or equal to 1. Therefore, if a/3 = 4/b, a/3 would also be less than or equal to 1. Multiplying both sides of the equation by 3 gives us a ≤ 3. So, statement (1) alone is sufficient to determine that a is less than or equal to 3. However, we cannot determine the relationship between a and b.

Statement (2): b ≤ 5

This statement tells us that b is less than or equal to 5. However, it does not provide any information about the value of a. Therefore, statement (2) alone is not sufficient to determine the relationship between a and b.

When we consider both statements together, we have the following information:

b ≥ 4 (from statement 1)
b ≤ 5 (from statement 2)

Combining these inequalities, we know that b must be between 4 and 5, inclusive. However, we still do not have enough information to determine the relationship between a and b.

Since statement (1) alone is sufficient to determine that a ≤ 3, but statement (2) alone is not sufficient, the correct answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): z < y

This statement tells us that z is less than y. However, it does not provide any information about the relationship between x and y. We cannot determine if x is less than y based on this statement alone.

Statement (2): z < x

This statement tells us that z is less than x. However, it does not provide any information about the relationship between x and y. We cannot determine if x is less than y based on this statement alone.

When we consider both statements together, we have the following information:

z < y (from statement 1)
z < x (from statement 2)

Combining these inequalities, we know that z is less than both y and x. However, we still do not have enough information to determine the relationship between x and y. It is possible that x is greater than y, or x is less than y. Therefore, even when both statements are considered together, we cannot determine if x is less than y.

Since neither statement alone, nor both statements together, provide sufficient information to answer the question, the correct answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To determine which statement(s) are sufficient to answer the question, we need to solve the inequalities given in each statement.
For statement (1), we have:
640y<320
Dividing both sides by 640, we get:
y<640320​=0.5
This tells us that y is less than 0.5, which is within the range of 0.3 to 0.6, but we don't know if it's greater than 0.3 based on this statement alone.
For statement (2), we have:
800y>320
Dividing both sides by 800, we get:
y>800320​=0.4
This tells us that y is greater than 0.4, which is also within the range of 0.3 to 0.6, but we don't know if it's less than 0.6 based on this statement alone.
Now, let's consider both statements together. From statement (1), we have y<0.5, and from statement (2), we have y>0.4. Combining these, we get:
0.4<y<0.5
This combined range is within the range of 0.3 to 0.6, so together, the statements are sufficient to answer the question. Neither statement alone is sufficient because each leaves part of the range 0.3 to 0.6 unchecked.
Therefore, the answer is:
3
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

FROM STATEMENT - I (INSUFFICIENT )

mn > np

Or, m > p

FROM STATEMENT - II (INSUFFICIENT )

mp > np

Or, m > n

FROM STATEMENT - I & II ( SUFFICIENT )

m > p & n

Hence, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked, answer will be (C)

To determine whether J/K>1, we need to evaluate the information provided by the two statements.
Statement (1): JK<1
This statement alone does not tell us whether J/K>1 because there are multiple combinations of positive numbers J and K whose product is less than 1. For example, if J=0.5 and K=2, then JK=1, but J/K=0.25, which is not greater than 1. Conversely, if J=0.5 and K=0.5, then JK=0.25, but J/K=1, which is equal to 1. Therefore, statement (1) alone is not sufficient.
Statement (2): J−K>0
This statement tells us that J is greater than K. If J is greater than K, and both are positive, then J/K would indeed be greater than 1. Therefore, statement (2) alone is sufficient to answer the question.
Based on this analysis, the answer is:
2
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(1) If 5 additional employees are hired by the company and all of the present employees remain, there will be at least 20 employees in the company.
E + 5 > = 20 = E > = 15 
With this we will not be able to exactly determine number of employees.
Hence, (1) ⇒ is NOT SUFFICIENT

(2) If no additional employees are hired by the company and 5 of the present employees resign, there will be fewer than 13 employees in the company.[b]
E − 5 < 13 = E < 18 
Once again this gives us a higher limit for E but still we cannot determine exact value of E
Hence, (2) ⇒ is NOT SUFFICIENT
Lets combine (1) & (2)
We get:
E >= 15
E < 18
E can be 15, 16, 17
As we are getting multiple values this is not sufficient.
(1) & (2) combined ⇒ is NOT SUFFICIENT
Hence, Answer is E

Statement (1): t is between 38 and 49.
From this statement, we can conclude that t is greater than or equal to 38 and less than or equal to 49. Since 38 is greater than 35 and 49 is less than 55, we can determine that t is indeed between 35 and 55. Statement (1) alone is sufficient.

Statement (2): t is between 42 and 52.
From this statement, we can conclude that t is greater than or equal to 42 and less than or equal to 52. Since 42 is greater than 35 and 52 is less than 55, we can determine that t is between 35 and 55. Statement (2) alone is also sufficient.

Therefore, each statement alone is sufficient to answer the question, and the correct answer is (D).