corvinis
If |x| > 3, which of the following must be true?
I. x > 3
II. x^2 > 9
III. |x - 1| > 2
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
If |x| > 3, then it must be true that
EITHER x > 3 OR x < -3(1) x > 3This need not be true, since it's also possible that x < -3.
For example, x COULD equal -5
So, statement I need not be true.
ELIMINATE answer choice A, C and E
Important: The remaining two answer choices (B and D) both state that statement II is true.
So, we need not analyze statement II, since we already know it must be true.
That said, let's analyze it for "fun"
(2) x² > 9This means that EITHER x > 3 OR x < -3
Perfect - this matches our original conclusion that
EITHER x > 3 OR x < -3(3) |x-1| > 2Let's solve this further.
We get two cases:
case a) x - 1 > 2, which means x > 3 PERFECT
or
case b) x - 1 < -2, which means x < -1
Must it be true that x < -1?
YES.
We already learned that
EITHER x > 3 OR x < -3If
x < -3, then we can be certain that x < -1
For example, if I tell you that the temperature is less than -3 degrees Celsius, can we be certain that the temperature is less than -1 degrees? Yes.
So, statement 3 must also be true.
Answer: D
Cheers,
Brent