Test: Inequalities - GMAT MCQ

Statement 1: 6 – 5x > -13
This statement provides an additional inequality. By solving this inequality, we can determine the range of values for x. Solving it:
6 - 5x > -13
Add 5x to both sides:
6 > -13 + 5x
19 > 5x
Divide both sides by 5 (since the inequality sign doesn't change when dividing by a positive number):
19/5 > x
x < 19/5

Statement 1 alone is sufficient to determine a range for x, but it doesn't provide an exact value for x.

Statement 2: 3 – 2x < -x + 4 < 7.2 – 2x
This statement provides a compound inequality. By solving this compound inequality, we can determine the range of values for x. Solving it:
3 - 2x < -x + 4 < 7.2 - 2x
We can simplify it by subtracting x from all parts of the inequality:
3 - 3x < 4 < 7.2 - 3x
Simplify further:
-3x + 3 < 4 < -3x + 7.2
Now we have two separate inequalities:
-3x + 3 < 4
4 < -3x + 7.2

Solving the first inequality:
-3x < 1
Divide by -3 (remember to reverse the inequality sign since we're dividing by a negative number):
x > -1/3

Solving the second inequality:
4 < -3x + 7.2
-3x < 3.2
Divide by -3 (reverse the inequality sign):
x > -3.2/3

Combining the two inequalities, we have:
x > -1/3 and x > -3.2/3

To find the common range of values for x, we take the greater of the two lower bounds, which is x > -1/3.

Statement 2 alone is sufficient to determine a range for x, but it doesn't provide an exact value for x.

Considering both statements together:
From statement 1, we know that x < 19/5, which gives us an upper bound for x.
From statement 2, we know that x > -1/3, which gives us a lower bound for x.

Combining the information, we have:
-1/3 < x < 19/5

Therefore, with both statements together, we have a range for x but not an exact value.

The answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: p⁴ > 256 This inequality states that the fourth power of p is greater than 256. To find the value of p, we need to take the fourth root of both sides of the inequality: ∛(p^4) > ∛256 p > 4

Statement 1 alone tells us that p is greater than 4. Therefore, it is sufficient to answer the question.

Statement 2: (p² - 1)(p - 4) > 0 This inequality is a quadratic expression. To determine its solution, we can analyze the sign of the expression for different values of p.

Consider the sign of each factor:

• (p² - 1) is positive for values of p outside the interval (-1, 1) and negative for values of p within (-1, 1).
• (p - 4) is positive for p > 4 and negative for p < 4.

To satisfy the inequality (p² - 1)(p - 4) > 0, we need both factors to have the same sign. This occurs when p > 4.

Statement 2 alone also tells us that p is greater than 4. Therefore, it is sufficient to answer the question.

Each statement alone is sufficient to answer the question and provides the same result: p > 4.

The answer is option D: EACH statement ALONE is sufficient to answer the question asked.

To determine if x > 0, we need to consider the discriminant of the quadratic equation x² + bx + c = 0.

The discriminant is given by Δ = b² - 4ac. If Δ > 0, the quadratic equation has two distinct real roots. If Δ = 0, it has one real root (a repeated root). If Δ < 0, it has no real roots.

Let's evaluate each statement:

Statement 1: b > 0
This statement alone does not provide information about the discriminant or the sign of x. It tells us that the coefficient b is positive, but we don't have enough information to determine the sign of x. Statement 1 is not sufficient.

Statement 2: c > 0
Similar to statement 1, this statement alone does not provide information about the discriminant or the sign of x. It tells us that the constant term c is positive, but we still cannot determine the sign of x. Statement 2 is not sufficient.

Considering both statements together:
We know that b > 0 and c > 0. However, even with this combined information, we still don't have enough information to determine the value of the discriminant Δ or the sign of x. It is possible to have cases where both b and c are positive, yet Δ < 0, leading to no real roots. Therefore, both statements together are not sufficient.

Since neither statement alone is sufficient and both statements together are also not sufficient, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement 1: n⁵(1 - n⁴) < 0
This inequality can be satisfied in two cases:
Case 1: n⁵ < 0 and 1 - n⁴ > 0
For n⁵ < 0, n must be negative, but for 1 - n⁴ > 0, n must be less than 1 or greater than -1. Therefore, n must be between -1 and 0.

Case 2: n⁵ > 0 and 1 - n⁴ < 0
For n⁵ > 0, n must be positive, but for 1 - n⁴ < 0, n must be greater than 1 or less than -1. Therefore, n must be outside the range of -1 to 0.

Statement 2: n⁴ - 1 < 0
This inequality can be satisfied when n⁴ < 1, which implies -1 < n < 1. Thus, n must be between -1 and 1.

Considering both statements together, we find that n must be between -1 and 0 according to statement 1, and between -1 and 1 according to statement 2. Therefore, the range of possible values for n is -1 < n < 0.

Since neither statement alone is sufficient to determine if n is negative, but together they provide a range for n, we conclude that both statements together are sufficient to answer the question. The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement 1: 0.25 < x < 0.35
This statement tells us that the value of x lies between 0.25 and 0.35. Comparing this range to the set {0.3, 0.22, 0.265}, we can see that x = 0.3 satisfies this condition. However, we cannot determine if any of the other values in the set satisfy this condition.

Statement 2: 0.2 < x < 0.28
This statement tells us that the value of x lies between 0.2 and 0.28. None of the values in the set {0.3, 0.22, 0.265} satisfy this condition.

Considering both statements together, we can conclude that the only value in the set that satisfies both conditions is x = 0.3. Therefore, both statements together are sufficient to answer the question. The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement 1: y² is less than y
This statement tells us that y² < y. From this inequality, we can deduce that y(1 - y) > 0. The solutions to this inequality are 0 < y < 1 or y > 1. However, since the question specifically asks if y is between 1 and 2, exclusive, the only possible value for y satisfying both conditions is y = 1. Therefore, statement 1 alone is sufficient to answer the question.

Statement 2: y² + y is between 1 and 2, exclusive
This statement tells us that 1 < y² + y < 2. By solving this inequality, we find that -2 < y < 1. However, this range does not satisfy the condition that y is between 1 and 2, exclusive. Therefore, statement 2 alone is not sufficient to answer the question.

Since statement 1 alone is sufficient to answer the question, and statement 2 alone is not, the answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: x - y = 2
This statement provides the difference between x and y, but it doesn't give us any specific information about the signs of x and y. For example, x could be 5 and y could be 3, in which case both x and y are positive. However, x could also be -1 and y could be -3, in which case both x and y are negative. Therefore, statement 1 alone is not sufficient to determine if x and y are both positive.

Statement 2: x/y > 1
This statement tells us that the ratio of x to y is greater than 1. If both x and y are positive, then their ratio will indeed be greater than 1. However, if x and y have opposite signs (one positive and one negative), their ratio could still be greater than 1. For example, if x = 2 and y = -1, then x/y = -2, which is greater than 1. Therefore, statement 2 alone is not sufficient to determine if x and y are both positive.

When we consider both statements together, we still cannot determine if x and y are both positive. The information provided in statement 1 doesn't give us any information about the signs of x and y, and statement 2 allows for the possibility of x and y having opposite signs. Therefore, both statements together are not sufficient to answer the question.

The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement 1: |a^b| > 0
This statement tells us that the absolute value of a raised to the power of b is greater than 0. Since a and b are integers and a not equal to b, we can deduce that a^b is not equal to 0. However, this statement does not provide any information about the sign of a^b. It is possible for a^b to be positive or negative, depending on the values of a and b. Therefore, statement 1 alone is not sufficient to answer the question.

Statement 2: |a|^b is a non-zero integer
This statement tells us that the absolute value of a raised to the power of b is a non-zero integer. Similarly to statement 1, this statement does not provide any information about the sign of a^b. It only tells us that the absolute value of a^b is a non-zero integer. It is still possible for a^b to be positive or negative. Hence, statement 2 alone is not sufficient to answer the question.

Since neither statement alone provides enough information to determine if |a|b is greater than 0, 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: xyz < 0
This statement tells us that the product of x, y, and z is negative. However, this statement does not provide any specific information about the relative positions of x, y, and z on the number line. It is possible that z lies between x and y, or it is also possible that z does not lie between x and y. Therefore, statement 1 alone is not sufficient to answer the question.

Statement 2: xy < 0
This statement tells us that the product of x and y is negative. Similar to statement 1, this statement does not provide any information about the position of z relative to x and y on the number line. It is also not sufficient to answer the question.

Since neither statement alone provides enough information to determine if z lies between x and y on the number line, 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: |x - 3| = 2x
This statement implies that the absolute value of the difference between x and 3 is equal to 2x. To solve this equation, we need to consider two cases:

Case 1: x - 3 = 2x
Solving this equation gives x = -3. However, we need to check if this solution satisfies the original equation.

|(-3) - 3| = 2(-3)
|-6| = -6 (which is not true)

Case 2: -(x - 3) = 2x
Simplifying this equation gives -x + 3 = 2x, which implies 3x = 3 and x = 1.

Therefore, statement 1 alone is sufficient, and we find that x = 1.

Statement 2: -(x + 3)² < 0
This statement tells us that the square of the expression -(x + 3) is less than zero. Since the square of any real number is non-negative, the only way for this inequality to hold is if -(x + 3) equals zero, which implies x = -3.

Therefore, statement 2 alone is not sufficient to determine the value of x.

Based on the evaluation of each statement, we conclude that statement 1 alone is sufficient to determine the value of x. The answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.