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