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29 Nov 2021, 04:43
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If \((|x| - 2)(x + 5) < 0\), then which of the following must be true?
A. \(x > 2\)
B. \(x < 2\)
C. \(-2 < x < 2\)
D. \(-5 < x < 2\)
E. \(x < -5\)
A. \(x > 2\)
B. \(x < 2\)
C. \(-2 < x < 2\)
D. \(-5 < x < 2\)
E. \(x < -5\)
29 Nov 2021, 04:43
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Official Solution:
If \((|x| - 2)(x + 5) < 0\), then which of the following must be true?
A. \(x > 2\)
B. \(x < 2\)
C. \(-2 < x < 2\)
D. \(-5 < x < 2\)
E. \(x < -5\)
\((|x| - 2)(x + 5) < 0\) means that \(|x| - 2\) and \(x + 5\) must have the opposite signs.
CASE 1: \(|x| - 2 > 0\) and \(x + 5 < 0\):
\(|x| - 2 > 0\) means that \(x < -2\) or \(x > 2\);
\(x + 5 < 0\) means that \(x < -5\).
Intersection of these ranges is \(x< -5\).
CASE 2: \(|x| - 2 < 0\) and \(x + 5 > 0\):
\(|x| - 2 < 0\) means that \(-2 < x < 2\);
\(x + 5 > 0\) means that \(x > -5\).
Intersection of these ranges is \(-2 < x < 2\).
So, we have that \((|x| - 2)(x + 5) < 0\) means that \(x < -5\) or \(-2 < x < 2\). ANY \(x\) from these possible ranges will for sure be less than 2 (option B).
To explaining other options:
\(x > 2\) (A) is not true because \(x\) could say be 0.
\(-2 < x < 2\) (C) is not true because \(x\) could say be -10.
\(-5 < x < 2\) (D) is not true because \(x\) could say be -10.
\(x < -5\) (E) is not true because \(x\) could say be 0.
Answer: B
If \((|x| - 2)(x + 5) < 0\), then which of the following must be true?
A. \(x > 2\)
B. \(x < 2\)
C. \(-2 < x < 2\)
D. \(-5 < x < 2\)
E. \(x < -5\)
\((|x| - 2)(x + 5) < 0\) means that \(|x| - 2\) and \(x + 5\) must have the opposite signs.
CASE 1: \(|x| - 2 > 0\) and \(x + 5 < 0\):
\(|x| - 2 > 0\) means that \(x < -2\) or \(x > 2\);
\(x + 5 < 0\) means that \(x < -5\).
Intersection of these ranges is \(x< -5\).
CASE 2: \(|x| - 2 < 0\) and \(x + 5 > 0\):
\(|x| - 2 < 0\) means that \(-2 < x < 2\);
\(x + 5 > 0\) means that \(x > -5\).
Intersection of these ranges is \(-2 < x < 2\).
So, we have that \((|x| - 2)(x + 5) < 0\) means that \(x < -5\) or \(-2 < x < 2\). ANY \(x\) from these possible ranges will for sure be less than 2 (option B).
To explaining other options:
\(x > 2\) (A) is not true because \(x\) could say be 0.
\(-2 < x < 2\) (C) is not true because \(x\) could say be -10.
\(-5 < x < 2\) (D) is not true because \(x\) could say be -10.
\(x < -5\) (E) is not true because \(x\) could say be 0.
Answer: B
