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
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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