Is |x−5|>4?
(1) x^2−4>0
(2) x^2−1<0
OA B
please elaborate . Thanks in advance .[spoiler][/spoiler]
Is |x−5|>4?
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}x-5| > 4?guerrero wrote:Is |x−5|>4?
(1) x^2−4>0
(2) x^2−1<0
OA B
please elaborate . Thanks in advance .[spoiler][/spoiler]
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.
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The question is asking whether the distance of x from 5 on the number line is greater than 4 or not.guerrero wrote:Is |x−5|>4?
(1) x^2−4>0
(2) x^2−1<0
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.
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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.
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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education