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Is |x – 5| < 5?

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Is |x – 5| < 5?

by gmattesttaker2 » Thu May 15, 2014 8:29 am
Hello,

For the following:

Is |x - 5| < 5?

(1) x < -4
(2) x < 4

OA: A


I first tried to simplify:

|x - 5| < 5 ?

Is, |x - 5| < 5 ?
=> + or - (x - 5) < 5?
=> (x - 5) < 5 or -(x - 5) < 5 ?
=> (x - 5) < 5 or (x - 5) > -5 ?
=> x-5+5 < 5+5 or x-5+5 > -5+5?
=> x < 10 or x > 0 ?

My question here was does this mean that x could be less than 10 or x could be greater than 0? (For instance x could be 20 since it is greater than 0) or does it mean that x has to be between 0 and 10?

Thanks a lot,
Sri
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by theCodeToGMAT » Thu May 15, 2014 8:41 am
To find: |x - 5 | < 5

In other words : is x > 0 & x <= 5 & x => 5 & < 10

Statement 1:
x < -4 ..
SUFFICIENT

Statement 2:
x < 4
X can be 1,2,3 or -1-2-3...
INSUFFICIENT

[spoiler]{A}[/spoiler]
Last edited by theCodeToGMAT on Thu May 15, 2014 8:49 am, edited 1 time in total.
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by ceilidh.erickson » Thu May 15, 2014 8:47 am
My question here was does this mean that x could be less than 10 or x could be greater than 0? (For instance x could be 20 since it is greater than 0) or does it mean that x has to be between 0 and 10?
Your first interpretation is correct. When you have a closed range (greater than a smaller number, less than a larger number), it makes more sense to combine into a 3-part inequality:
0 < x < 10
In other words: is x between 0 and 10?

If you had the opposite (greater than a larger number, less than a smaller number), as in |x - 5| > 5, you would interpret this as:
x < 0 or x > 10
Since those are two non-overlapping ranges, we have to think in terms of "or."


Is |x - 5| < 5?
Rephrase: Is x between 0 and 10?

(1) x < -4
Sufficient. If x is less than -4, it's definitely not between 0 and 10, so we have a definitive "no" answer to the question.

(2) x < 4
Insufficient. Some values less than 4 are between 0 and 10; some are not.
Ceilidh Erickson
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Harvard Graduate School of Education
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by GMATGuruNY » Thu May 15, 2014 12:23 pm
gmattesttaker2 wrote:Hello,

For the following:

Is |x - 5| < 5?

(1) x < -4
(2) x < 4
|a-b| = the DISTANCE between a and b on the number line.

|x-5| = the distance between x and 5 on the number line.
Question stem, rephrased:
On the number line, is the distance between x and 5 less than 5?

Since 0 is 5 places to the left of 5, while 10 is 5 places to the right of 5, we need to determine whether x is within the following range in red:
0<----less than 5 places----5----less than 5 places---->10

Statement 1: x<-4
Since x is negative, it cannot be within the required range.
SUFFICIENT.

Statement 2: x<4
It's possible that x=3, which is within the required range.
It's possible that x=-1, which is NOT within the required range.
INSUFFICIENT.

The correct answer is A.
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by [email protected] » Thu May 15, 2014 7:00 pm
Hi Sri,

You could leave the question as is and TEST VALUES.

Is |X - 5| < 5? This is a YES/NO question.

Fact 1: X < -4

From Fact 1, we know that X is "more negative" than -4, so it could be -5, -6, -7, etc. X could also be a non-integer, such -4.01.

If X = -5, then |-5 - 5| = 10 < 5? The answer is NO.
If X = -6, then |-6 - 5| = 11 < 5? The answer is NO.
If X = -4.01, then |-4.01 - 5| = 9.01 < 5? The answer is NO.
Fact 1 is SUFFICIENT

Fact 2: X < 4

Notice how all of the values that "fit" Fact 1 also fit Fact 2?
If X = -5, the answer is NO.

If X = 3, then |3 - 5| = 2 < 5? The answer is YES.
Fact 2 INSUFFICIENT.

Final Answer: A

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