Test: Absolute Values/Modules - GMAT MCQ

To determine if r²/∣r∣ ​< 1, we can analyze the given statements:

Statement (1): r > −1
Statement (2): r < 1

Let's evaluate each statement separately:

Statement (1) states that r is greater than -1. Since the square of any positive number is always greater than or equal to the original number itself, we can conclude that r² ≥ r. Additionally, since r is positive, ∣r∣ = r. Therefore, So, statement (1) alone is sufficient to answer the question.

Statement (2) states that r is less than 1. Again, since r is positive, ∣r∣ = r. Thus, Therefore, statement (2) alone is also sufficient to answer the question.

Based on the analysis of each statement, both statements individually yield the same result: r² ≥ r. Consequently, statements (1) and (2) together are sufficient to answer the question.

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

Let's analyze the given statements:

(1) x = |x| (2) x² = 9

Statement (1) tells us that x is equal to the absolute value of x. This means that x can either be a positive number or zero. However, we don't know the exact value of x from this statement alone.

Statement (2) tells us that x squared is equal to 9. Taking the square root of both sides, we find two possible solutions: x = 3 or x = -3.

Now, let's consider both statements together. From statement (1), we know that x is either positive or zero. From statement (2), we have two possible values for x: x = 3 or x = -3. Since 0 is the only value that satisfies both statements, we can conclude that |x| = 0.

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

Let's analyze the given statements:

(1) |x + 6| > 6
(2) |x - 6| > 6

Statement (1) tells us that the absolute value of x plus 6 is greater than 6. This means that the distance between x and -6 is greater than 6. We can interpret this as x being either greater than 0 or less than -12. However, we cannot determine if x is positive from this statement alone.

Statement (2) tells us that the absolute value of x minus 6 is greater than 6. This means that the distance between x and 6 is greater than 6. We can interpret this as x being either greater than 12 or less than 0. However, we cannot determine if x is positive from this statement alone either.

By combining both statements, we find that x must satisfy both conditions simultaneously. From statement (1), we know that x can be greater than 0 or less than -12. From statement (2), we know that x can be greater than 12 or less than 0. The only range that satisfies both conditions is x being less than -12.

Therefore, neither statement alone is sufficient to determine if x is positive. The correct answer is E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Let's analyze the given statements:

(1) x + y = 20
(2) |x + y| = 20

Statement (1) tells us that the sum of x and y is equal to 20. However, it doesn't provide any specific information about the values of x and y individually. We cannot determine the average of x and |y| from this statement alone.

Statement (2) tells us that the absolute value of the sum of x and y is equal to 20. This means that the sum of x and y can either be 20 or -20. However, we still cannot determine the values of x and y individually or their average from this statement alone.

By combining both statements, we have the equation x + y = 20 from statement (1) and |x + y| = 20 from statement (2). Since the absolute value of a number can only be positive or zero, we can conclude that x + y must be either 20 or -20.

However, we still don't have enough information to determine the values of x and y individually or their average. We don't know how the sum is distributed between x and y. For example, x could be 10 and y could be 10, which would result in an average of 10. Or x could be 15 and y could be 5, which would also result in an average of 10. There are multiple possibilities, and additional data is needed to determine the average of x and |y|.

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

Statement (1): 3r > 3s

This statement alone doesn't provide any information about t or the relationship between t, r, and s. It only tells us that 3r is greater than 3s, which doesn't help us determine if t = |r - s|.

Statement (2): t = r - s

This statement directly tells us the relationship between t, r, and s. It states that t is equal to the difference between r and s.

Now, let's consider both statements together.

Combining statement (1) and statement (2), we can rewrite statement (1) as: r > s.

If r > s, then r - s will be a positive value. Taking the absolute value of a positive value will still result in the same positive value.

Therefore, t = |r - s| is true when r > s, which is implied by statement (1).

Based on this analysis, we can conclude that BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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

Statement (1): |x + 9| = y

This statement tells us that the absolute value of (x + 9) is equal to y. However, without any information about the value of x, we cannot determine the specific value of y. Therefore, statement (1) alone is not sufficient.

Statement (2): y² = 9

This statement tells us that y squared is equal to 9. Taking the square root of both sides, we have two possible values for y: y = 3 or y = -3. Hence, statement (2) alone is sufficient to determine the possible values of y.

When we consider both statements together, we have additional information. From statement (2), we know that y can be either 3 or -3. However, statement (1) does not provide any specific information about the value of x or restrict the possible values of y further.

Therefore, when considering both statements together, we have multiple possible values for y (3 and -3), and we cannot determine a unique value. Hence, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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

To determine the value of x when |x| < 4, let's evaluate each statement:

Statement (1): x is an integer divisible by 3.

Since |x| < 4, the possible values of x are -3, -2, -1, 0, 1, 2, or 3. However, statement (1) tells us that x must be divisible by 3, which narrows down the possibilities to -3, 0, or 3.

Statement (2): x is an integer divisible by 2.

Similar to statement (1), the possible values of x are -3, -2, -1, 0, 1, 2, or 3. However, statement (2) tells us that x must be divisible by 2, which narrows down the possibilities to -2, 0, or 2.

Combining both statements, we see that the only possible value of x that satisfies both conditions is 0.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient. The answer is (C).

Statement (1): |x² + 16| - 5 = 27

Adding 5 to both sides, we have |x² + 16| = 32.

Considering the absolute value, we can rewrite the equation as two separate cases:

x² + 16 = 32:
Solving this equation gives us x² = 16, which means x = ±4. Therefore, |x| = 4.

-(x² + 16) = 32:
Simplifying, we have -x² - 16 = 32.
Rearranging, we get -x² = 48, and dividing by -1, we have x² = -48. Since the square of any real number is non-negative, there is no real solution for x in this case.

Statement (1) alone gives us two possible values for |x|: 4 or no solution.

Statement (2): x² = 8x - 16

Rearranging, we get x² - 8x + 16 = 0. This equation can be factored as (x - 4)(x - 4) = 0, which means x = 4.

Statement (2) alone gives us the value x = 4, which means |x| = 4.

Considering both statements together, we see that they both give the same value for |x|, which is 4.

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

To determine if |x - y| > 0, let's analyze the given equation xy + z = z:

If we subtract z from both sides of the equation, we get xy = 0. This equation implies that at least one of the variables x or y must be zero.

Now let's evaluate the statements:

Statement (1): x ≠ 0
If x ≠ 0, it means x is not zero. Since xy = 0, this implies that y must be zero. Therefore, |x - y| = |x - 0| = |x| = |x| > 0. Statement (1) alone is sufficient to answer the question.

Statement (2): y = 0
If y = 0, then xy = 0. This implies that x can be any value, including zero. In this case, |x - y| = |x - 0| = |x| = |0| = 0. Statement (2) alone is not sufficient to answer the question.

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

Statement (1): |x + 9| = y

This statement tells us that the absolute value of (x + 9) is equal to y. However, without any information about the value of x, we cannot determine the specific value of y. Therefore, statement (1) alone is not sufficient.

Statement (2): y² = 9

This statement tells us that y squared is equal to 9. Taking the square root of both sides, we have two possible values for y: y = 3 or y = -3. Hence, statement (2) alone is sufficient to determine the possible values of y.

When we consider both statements together, we have additional information. From statement (2), we know that y can be either 3 or -3. However, statement (1) does not provide any specific information about the value of x or restrict the possible values of y further.

Therefore, when considering both statements together, we have multiple possible values for y (3 and -3), and we cannot determine a unique value. Hence, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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