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20 May 2020, 06:54
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
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Question Stats:
31% (02:13) correct
69%
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wrong
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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
Are You Up For the Challenge: 700 Level Questions
M37-58
A. x > 2
B. x < 2
C. -2 < x < 2
D. -5 < x < 2
E. x < -5
Are You Up For the Challenge: 700 Level Questions
M37-58
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29 Nov 2021, 04:44
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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
General Discussion
20 May 2020, 07:03
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Bunuel
(|x| - 2)(x + 5) < 0
(I) Make use of choices.
a) If x=0, (|x| - 2)(x + 5) < 0...... => (0 - 2)(0 + 5) < 0....-10<0...YES
So the range should be containing 0..
Eliminate A and E
a) If x=-10, (|x| - 2)(x + 5) < 0...... => (10 - 2)(-10 + 5) < 0....-40<0...YES
So the range should be containing -10..
Eliminate C and D
B
(II) Algebraic way
\((|x| - 2)(x + 5) < 0\)
If (|x| - 2)>0 or |x|>2, that is x<-2 and x>2, then (x + 5) < 0 or x<-5.....Common range x<-5
If (|x| - 2)<0 or |x|<2, that is -2<x<2, then (x + 5) > 0 or x>-5.....Common range -2<x<2
Combined range...x<-5 and -2<x<2
Look at the choices that contain both these ranges....x<2
B
NOTE : Do not get confused that -5<x<-2 does not satisfy the inequality, we are looking at what must be true and NOT the range of x.
Even if it was x<100, it would be correct as an answer as it would contain both the ranges x<-5 and -2<x<2
