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Test: Inequalities - GMAT MCQ


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10 Questions MCQ Test - Test: Inequalities

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Test: Inequalities - Question 1

If 8x > 4 + 6x, what is the value of the integer x?

(1) 6 – 5x > -13
(2) 3 – 2x < -x + 4 < 7.2 – 2x

Detailed Solution for Test: Inequalities - Question 1

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.

Test: Inequalities - Question 2

If p is a positive integer, is p > 4 ?

(1) p4 > 256
(2) (p2 -1)(p - 4) > 0

Detailed Solution for Test: Inequalities - Question 2

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.

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Test: Inequalities - Question 3

For integers x, b and c, x2 + bx + c = 0. Is x > 0?

(1) b > 0
(2) c > 0

Detailed Solution for Test: Inequalities - Question 3

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.

Test: Inequalities - Question 4

Is n negative?

(1) n5(1 - n4) < 0
(2) n4 - 1 < 0

Detailed Solution for Test: Inequalities - Question 4

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.

Test: Inequalities - Question 5

If x is a member of the set {0.3, 0.22, 0.265}, what is the value of x?

(1) 0.25 < x < 0.35
(2) 0.2 < x < 0.28

Detailed Solution for Test: Inequalities - Question 5

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.

Test: Inequalities - Question 6

Is y between 1 and 2, exclusive?

(1) y2 is less than y
(2) y2 + y is between 1 and 2, exclusive

Detailed Solution for Test: Inequalities - Question 6

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.

Test: Inequalities - Question 7

Are x and y both positive?

(1) x - y = 2
(2) x/y > 1

Detailed Solution for Test: Inequalities - Question 7

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.

Test: Inequalities - Question 8

If a and b are integers, and a not= b, is |a|b > 0?

(1) |ab| > 0
(2) |a|b is a non-zero integer

Detailed Solution for Test: Inequalities - Question 8

Statement 1: |ab| > 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 ab is not equal to 0. However, this statement does not provide any information about the sign of ab. It is possible for ab 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 ab. It only tells us that the absolute value of ab is a non-zero integer. It is still possible for ab 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.

Test: Inequalities - Question 9

On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?

(1) xyz < 0
(2) xy < 0

Detailed Solution for Test: Inequalities - Question 9

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.

Test: Inequalities - Question 10

If x is an integer, what is the value of x?

(1) |x – 3| = 2x
(2) –(x + 3)^2 < 0

Detailed Solution for Test: Inequalities - Question 10

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.

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