Is |x−5|>4?

[guerrero]

Is |x−5|>4?

(1) x^2−4>0

(2) x^2−1<0

OA B


please elaborate . Thanks in advance .[spoiler][/spoiler]

[GMATGuruNY]

Is |x−5|>4?

(1) x^2−4>0

(2) x^2−1<0

OA B


please elaborate . Thanks in advance .[spoiler][/spoiler]

}x-5| > 4?

Case 1: x-5 > 4
x > 9.

Case 2: -(x-5) > 4
-x + 5 > 4
-x > -1
x < 1.

Thus:
|x-5| > 4 if x<1 or x>9.
Put another way:
|x-5| ≤ 4 if 1≤x≤9.

Question rephrased: Is 1≤x≤9?

Statement 1: x² > 4.
If x=3, then 1≤x≤9.
If x=10, then x>9.
INSUFFICIENT.

Statement 2: x² < 1.
No value between 1 and 9, inclusive, will work here.
Thus, it is not possible that 1≤x≤9.
SUFFICIENT.

The correct answer is B.

[Anurag@Gurome]

Is |x−5|>4?

(1) x^2−4>0
(2) x^2−1<0

The question is asking whether the distance of x from 5 on the number line is greater than 4 or not.

Statement 1: x² - 4 > 0 ---> x > 2 or x < -2
If x = 5 ---> |x - 5| = 0 < 4
If x = 10 ---> |x - 5| = 5 > 4

Not sufficient

Statement 2: x² - 1 < 0 ---> -1 < x < 1
As x is less than 1, distance of x from 5 on the number line will be always greater than 4.

Sufficient

The correct answer is B.

[ceilidh.erickson]

This question CANNOT be an official GMAT question - the statements contradict each other! Beware of studying from GMAT sources that do not conform to GMAT rules!

But before we get to that, let's analyze how we would approach a legal absolute value / inequality question...

When looking at absolute values with inequalities, I find it most helpful to think in terms of a number line. Where will (x - 5) have a distance from 0 of more than 4?



Our real question: is x > 9 or x < -1 ?

(1) x^2 > 4

Be careful with squares and inequalities! We need to think about the positive and negative case, and thus flipping the signs. Think about this on a number line:



As we can see, there are values that will give us a "yes" answer to our question (e.g. 10), and values that give us a "no" answer (e.g. 5). Insufficient.

(2) x^2 < 1
Again, let's see this on a number line:



As we can see, this entire range satisfies x < 1, so it's Sufficient.

HOWEVER!!!... on a real GMAT question, the statements can't possibly contradict each other, because they have to both be TRUE. It's not possible for x^2 to be both greater than 4 and less than 1. These statements contradict each other, therefore this is an impossible question.