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Difficulty:
45%
(medium)
Question Stats:
44% (00:51) correct
55% (00:48) wrong based on 29 sessions
History
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If ab = |a| × |b| and ab ≠ 0, which of the following must be true?
Indicate all such statements.
A. a = b
B. a > 0 and b > 0
C. ab > 0
Indicate all such statements.
A. a = b
B. a > 0 and b > 0
C. ab > 0
09 Nov 2025, 03:34
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Explanation
\(ab=∣ab∣.\) For any real number, \(x=∣x∣\) means \(x≥0\). Here \(x=ab.\) So \(ab≥0.\) But \(ab ≠ 0\) (given), so \(ab>0.\) It means \(a\) and \(b\) have the same sign (both positive or both negative).
Check each option
A. \(a=b\) Not necessarily. Example: \(a=2,b=3 → ab=6, ∣a∣∣b∣=6,\) works, but \(a≠b.\) So false.
B. \(a>0\) and \(b>0\) Not necessarily. Example: \(a=−2,b=−3 → ab=6, ∣a∣∣b∣=6,\) works, but \(a\) and \(b\) are not both \(> 0.\) So false.
C. \(ab>0\) Yes, we deduced \(ab>0\) from the given conditions. So true.
Answer: C
\(ab=∣ab∣.\) For any real number, \(x=∣x∣\) means \(x≥0\). Here \(x=ab.\) So \(ab≥0.\) But \(ab ≠ 0\) (given), so \(ab>0.\) It means \(a\) and \(b\) have the same sign (both positive or both negative).
Check each option
A. \(a=b\) Not necessarily. Example: \(a=2,b=3 → ab=6, ∣a∣∣b∣=6,\) works, but \(a≠b.\) So false.
B. \(a>0\) and \(b>0\) Not necessarily. Example: \(a=−2,b=−3 → ab=6, ∣a∣∣b∣=6,\) works, but \(a\) and \(b\) are not both \(> 0.\) So false.
C. \(ab>0\) Yes, we deduced \(ab>0\) from the given conditions. So true.
Answer: C
