If |x - 2| < 5, which of the following must be true?

|x - 2| < 5
can be evaluated as "Distance of x from 2 must be less than 5."
Points at a distance of 5 from 2 are -3 and 7.
So x must be greater than -3 but less than 7.

So x can take values such as -2.5, -1, 0.46, 1.4, 5, 6 etc. Note that each of these values WILL be greater than -4.

Answer (C)
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In reference to a PM.
Some range MUST be true means that all the possible values should be within the option. It may be spread much more.
For example
X is an even number.
What must be true ? :- X must be an integer.

Here we have |x-2|<5
Two cases
1) x>2
x-2<5....x<7
2) x<2
2-x<5......x>-3

Which options has values -3 to 7

Only C, where x>-4
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I'm weirded out about the x > -4 answer. This answer states that x can be any value bigger than -3, so also e.g. x can be 10. But this doesn't suit the original constraint as in 10 > 7.

Please explain

X cannot be 10 because it does not fall with in the range given in the question stem i.e. \( -3<x<7\)
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Why cant the ans be D, x definitely falls between 0 to 3, x>-4 can must mean 10, but if we pick any value between 0 and 3, it always falls between -3 to 7. Experts reply

Hi DonZepp and PgRaul,

To start, any time an Absolute Value appears in a Quant question, while there will likely be some solutions to the given equation/inequality that are 'obvious', there will almost certainly be some OTHER solutions that you also have to consider.

With this prompt, we're told that |X - 2| < 5 and we're asked which of the following MUST be true.

The 'obvious' solutions to this inequality are positive integers (re: X = 6, 5, 4, 3 and 2), but there are other solutions, including 1, 0, -1 and -2. Since we're dealing with an inequality, we do have to consider the 'upper limit' and 'lower limit' of the range of solutions, so...

-3 < X < 7

Thus, ANY value of X in that range is a solution for this inequality. Considering THAT 'limitation', which of the following answers is ALWAYS true?

A) X>0.... does X HAVE to be greater than 0? NO... there are a bunch of negative numbers that are solutions.
B) X>8.... does X HAVE to be greater than 8? NO... that's just fundamentally incorrect.
C) X>-4.... does X HAVE to be greater than -4? YES... every possible solution to this inequality IS greater than -4, so this answer IS ALWAYS true. This answer does NOT mean that every number that is greater than -4 is a solution to this inequality. It IS always true based on the set of solutions though (and that's what the question was asking for).

Final Answer:

GMAT assassins aren't born, they're made,
Rich

I dont know, but still I am not convinced with this explanation. Because -3<x<7 is the original constraint after solving the equation. So if we consider x>-4, then any value greater than -4 should full fill the constraint. But if we consider any value greater than 7, that is not full filling the constraint. So x>-4 is not always true. But if we take 0<x<3, then any values will full fill the constraint. So this option is always true.

Oh okay. I understood. So if we choose a value between -3 to 7, it will always be greater than -4, not the other way round of we choosing a value which is greater than -4. so option C is correct. Option D is wrong, because if we choose a value between -3 to 7, it shouldn't necessarily fall in the range 0<x<3. Thank you

Hi sampad,

The 'logic' that this question is based around is a relatively rare 'design' (and you won't necessarily see it on Test Day). It might help to think in 'broad' terms of what we know is true.

For example, is the number 1 greater than 0? Yes.
Is 1 greater than -1? Yes.
Is 1 greater than -918? Yes.

It might seem a little strange to 'compare' the number 1 to the number -918, but the fact that 1 is greater than -918 is still a true statement. The same idea is what this prompt is built around. We have a range of numbers (re: - 3 < X < 7). Which of the answer choices is true for ALL of the values of X in THAT range....?

GMAT assassins aren't born, they're made,
Rich

for option c) X>-4, X can also be -3.5 and it does not satisfy the statement
so why did we assume that X is an integer?

help appreciated.

Hi priyabaheti,

With this prompt, we're told that |X - 2| < 5 and we're asked which of the following MUST be true. We are not told that X is an integer.

While the integer solutions for X are 6, 5, 4, 3, 2, 1, 0, -1 and -2, since we're dealing with an inequality, we do have to consider the actual 'upper limit' and 'lower limit' of the range of solutions. They are...

-3 < X < 7

THIS is what the prompt tells us about X could be (re: any value from -3 to 7, but not -3 or 7). Considering THAT 'limitation', which of the following answers is ALWAYS true?

A) X>0.... does X HAVE to be greater than 0? NO... there are a bunch of negative numbers that are solutions.
B) X>8.... does X HAVE to be greater than 8? NO... that's just fundamentally incorrect.
C) X>-4.... does X HAVE to be greater than -4? YES... every possible solution to this inequality IS greater than -4, so this answer IS ALWAYS true. This answer does NOT mean that every number that is greater than -4 is a solution to this inequality. It IS always true based on the set of solutions though (and that's what the question was asking for).

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com

Absolute value means distance. x-2 can't be farther than 5 units away from 0. That means that x-2 has to be less than 5 AND x-2 has to be greater than -5.

x-2 < 5
x < 7

AND

x-2 > -5
x >-3

If we graph that, the portion of the number line that we color in is from -3 through 7.

A) Does x HAVE to be greater than 0? No, it could be negative. Eliminate.
B) Does x HAVE to be greater than 8? No, it can't be greater than 8. Eliminate.
C) Does x HAVE to be greater than -4? Yes, there is no value less than -4 colored in on the number line.

Answer choice C.

I don't know this guy, but here are two good videos for learning how to handle absolute value inequalities. Make sure you understand the difference that he talks about between AND and OR.

I contemplate that this is the wrong question.
(x-2)^2 < 25
x^2 - 4x - 21 < 0
x(1) = 7
x(2) = -3
Thereby, the equation's solution will cover the range from -3 to 7. Hence, the most proper answer should be D.

You’ve said that x can be -2 or 6. Are those included in the range defined by D? Are they included in the range defined by C?

Posted from my mobile device

Hey,

I would appreciate a clarification - Why is the correct answer B and not E? the question does not states that X is an integer....

According to answer B, X can get any value greater than (-4). However, if X get the value 3.5 it does not satisfy the equation. Therefore not always true.

Am I missing something? Why did we assume that X is an integer?

Thank you

I am sure, you meant C while writing B.

Put any value that could be answer in the option C. If you get ananswer yes everytime, then C is true.

Say x is positive integer, then what should be true?
A. x is an integer
Will the above be true? Yes, it will be. 
Same logic as in the question­
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If |x - 2| < 5, which of the following must be true?

(A) x > 0
(B) x > 8
(C) x > -4
(D) 0 < x < 3
(E) None of the above­­

|x - 2| < 5

-5 < x - 2 < 5

-3 < x < 7

Any x from this range satisfies only one inequality from the given options, namely D, x > -4. Since -3 < x < 7 is true, then x must be greater than -4.

Answer: D.

Check other similar questions from Trickiest Inequality Questions Type: Confusing Ranges (part of our Special Questions Directory).

Hope it helps.­­
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