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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If 7<x^2-4x+12, which of the following MUST be true?
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28 Oct 2019, 09:40
28 Oct 2019, 09:40
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Question Stats:
27% (01:46) correct
72% (02:23) wrong based on 110 sessions
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If \(7<x^2-4x+12\), which of the following MUST be true?
i) \(3^{x-2}>0\)
ii) \(|x^3+1|>0\)
iii) \(\sqrt{(x+2)^2}>0\)
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i, ii and iii
i) \(3^{x-2}>0\)
ii) \(|x^3+1|>0\)
iii) \(\sqrt{(x+2)^2}>0\)
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i, ii and iii
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If 7<x^2-4x+12, which of the following MUST be true?
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28 Oct 2019, 11:37
28 Oct 2019, 11:37
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GreenlightTestPrep wrote:
If \(7<x^2-4x+12\), which of the following MUST be true?
i) \(3^{x-2}>0\)
ii) \(|x^3+1|>0\)
iii) \(\sqrt{(x+2)^2}>0\)
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i, ii and iii
i) \(3^{x-2}>0\)
ii) \(|x^3+1|>0\)
iii) \(\sqrt{(x+2)^2}>0\)
A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i, ii and iii
APPROACH #1: Start with the given inequality and then check the statements
Given: \(7<x^2-4x+12\)
Subtract 8 from both sides to get: \(-1<x^2-4x+4\)
Factor right side to get: \(-1<(x-2)^2\)
Since \((x-2)^2\) is ALWAYS greater than or equal to zero, we can see that the inequality is true for ALL values of x.
In other words, x can equal ANY number
i) \(3^{x-2}>0\)
This is true for all values of x.
So, statement i is MUST be true
ii) \(|x^3+1|>0\)
This inequality is NOT satisfied when \(x = -1\)
Since x can equal -1, statement ii need not be true
iii) \(\sqrt{(x+2)^2}>0\)
This inequality is NOT satisfied when \(x = -2\)
Since x can equal -2, statement iii need not be true
Answer: A
--------------------------------------
APPROACH #2: Start with the statements and then check the given inequality
i) \(3^{x-2}>0\)
This is true for all values of x.
So, statement i is MUST be true
ii) \(|x^3+1|>0\)
This inequality is NOT satisfied when \(x = -1\)
Check the given inequality to see if x CAN equal -1
Plug in x= -1 to get: \(7<(-1)^2-4(-1)+12\)
Simplify: \(7<17\). WORKS.
Since x can equal -1, statement ii need not be true
iii) \(\sqrt{(x+2)^2}>0\)
This inequality is NOT satisfied when \(x = -2\)
Plug in x= -2 to get: \(7<(-2)^2-4(-2)+12\)
Simplify: \(7<24\). WORKS.
Since x can equal -2, statement iii need not be true
Answer: A
Cheers,
Brent
If 7<x^2-4x+12, which of the following MUST be true?
[#permalink]
25 May 2022, 15:40
25 May 2022, 15:40
1
I did it with a bit different solutions. Do you see anything problems with my way?
x^2 - 4x - 5 > 0
Add 9 to both sides
x^2 - 4x + 4 > 9
Use the equation
x^2 - 4x + 4 = (x - 2)^2
Combines with the original
(x - 2)^2 > 9
Take a root and get two possible solutions
x - 2 > 3
-(x - 2) > 3
Therefore
x > 5
x < -1
A - Will be always correct
B - Cannot be correct because x cannot be -1
C - It cannot be equal to 2
x^2 - 4x - 5 > 0
Add 9 to both sides
x^2 - 4x + 4 > 9
Use the equation
x^2 - 4x + 4 = (x - 2)^2
Combines with the original
(x - 2)^2 > 9
Take a root and get two possible solutions
x - 2 > 3
-(x - 2) > 3
Therefore
x > 5
x < -1
A - Will be always correct
B - Cannot be correct because x cannot be -1
C - It cannot be equal to 2
