VeritasPrepKarishma wrote:
corvinis wrote:
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
I can't understand how the official answer can be right. For me is B. Please respond and I'll provide official explanation! Thanks
Responding to a pm:
|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3
Which of the following must be true?
I. x > 3
For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.
II. X^2 > 9
This is the same as |x| > 3 so it must be true
III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1
For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.
Responding to a pm:
Quote:
I couldn't understand the solution for option B.
Since |x|>3 we can say that |x|-1>2 ( subtracting 2 from both sides).
But how are we saying that |x|-1 is equal to |x-1|.
They are not the same: |x - 1| > 2 and |x|-1 > 2
|x - 1| > 2
means x > 3 or x < -1
|x| - 1 > 2
|x| > 3
means x > 3 or x < -3
But not what is given and what is asked.
We are GIVEN that |x| > 3
So we KNOW that x is either greater than 3 or it is less than -3. So valid values for x are 3.4, 4, 101, 2398675, -3.6, -5, -78 etc
Now the question is:
"Is |x - 1| > 2?"
"Is x always either greater than 3 or less than -1?"
All positive values of x are given to be greater than 3.
All negative values of x are given to be less than -3. So obviously they are less than -1 too.
Hence, |x - 1| > 2 is true.
Helps?