GMAT Exam > GMAT Questions > If |a| = |b|, which of the following must be ...
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If |a| = |b|, which of the following must be true?
I. a = b
II. |a| = -b
III. -a = -b
I. a = b
II. |a| = -b
III. -a = -b
- a)I only
- b)II only
- c)III only
- d)I and III only
- e)None
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
If |a| = |b|, which of the following must be true?I. a = bII. |a| = -b...
Because we know that |a| = |b|, we know that a and b are equidistant from zero on the number line. But we do not know anything about the signs of a and b (that is, whether they are positive or negative). Because the question asks us which statement(s) MUST be true, we can eliminate any statement that is not always true. To prove that a statement is not always true, we need to find values for a and b for which the statement is false.
I. NOT ALWAYS TRUE: a does not necessarily have to equal b. For example, if a = -3 and b = 3, then |-3| = |3| but -3 ≠ 3.
II. NOT ALWAYS TRUE: |a| does not necessarily have to equal -b. For example, if a = 3 and b = 3, then |3| = |3| but |3| ≠ -3.
III. NOT ALWAYS TRUE: -a does not necessarily have to equal -b. For example, if a = -3 and b = 3, then |-3| = |3| but -(-3) ≠ -3.
The correct answer is E.
I. NOT ALWAYS TRUE: a does not necessarily have to equal b. For example, if a = -3 and b = 3, then |-3| = |3| but -3 ≠ 3.
II. NOT ALWAYS TRUE: |a| does not necessarily have to equal -b. For example, if a = 3 and b = 3, then |3| = |3| but |3| ≠ -3.
III. NOT ALWAYS TRUE: -a does not necessarily have to equal -b. For example, if a = -3 and b = 3, then |-3| = |3| but -(-3) ≠ -3.
The correct answer is E.
Most Upvoted Answer
If |a| = |b|, which of the following must be true?I. a = bII. |a| = -b...
Explanation:
Given:
|a| = |b|
Analysis:
When |a| = |b|, it means that the absolute values of a and b are equal. However, the signs of a and b could be different, leading to different possible relationships between a and b.
Options Analysis:
I. a = b - This is not necessarily true. For example, a could be 5 and b could be -5, where |a| = |b| but a does not equal b.
II. |a| = -b - This is not possible, as the absolute value of a is always non-negative, while -b is always negative.
III. -a = -b - This is also not necessarily true. For example, a could be 3 and b could be -3, where |a| = |b| but -a does not equal -b.
Conclusion:
None of the given statements must be true when |a| = |b|. The relationship between a and b can vary, so option E, "None," is the correct answer.
Given:
|a| = |b|
Analysis:
When |a| = |b|, it means that the absolute values of a and b are equal. However, the signs of a and b could be different, leading to different possible relationships between a and b.
Options Analysis:
I. a = b - This is not necessarily true. For example, a could be 5 and b could be -5, where |a| = |b| but a does not equal b.
II. |a| = -b - This is not possible, as the absolute value of a is always non-negative, while -b is always negative.
III. -a = -b - This is also not necessarily true. For example, a could be 3 and b could be -3, where |a| = |b| but -a does not equal -b.
Conclusion:
None of the given statements must be true when |a| = |b|. The relationship between a and b can vary, so option E, "None," is the correct answer.
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