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# How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

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GMAT Forum Moderator
Joined: 28 May 2014
Posts: 528
GMAT 1: 730 Q49 V41
How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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23 Mar 2017, 00:02
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How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

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Math Expert
Joined: 02 Sep 2009
Posts: 46207
Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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23 Mar 2017, 00:11
saswata4s wrote:
How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

|x - 5| ≤ 2.5;

Get rid of modulus: -2.5 ≤ x - 5 ≤ 2.5;

Add 5 to all three parts: 2.5 ≤ x ≤ 7.5

Integer values of x that satisfy the inequality are 3, 4, 5, 6, and 7. So, 5 values.

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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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23 Mar 2017, 03:05
Top Contributor
Hi saswata4s,

We can solve this question using number line too, because modulus (or absolute value) is nothing but the length,

Here the question says,

|x-5| ≤ 2.5

That is distance between x and 5 has to be less than or equal to 2.5,

So, let’s draw this in the number line, we can see that from the below diagram, that

x has to be between 2.5 and 7.5,

The values are 3,4,5,6 and 7.

So there are 5 values,

Hope this helps.
Attachments

number line-GC.png [ 12.15 KiB | Viewed 3906 times ]

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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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23 Mar 2017, 03:31
saswata4s wrote:
How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

This post discusses this concept in detail:
https://www.veritasprep.com/blog/2011/0 ... edore-did/
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 22 Oct 2016 Posts: 22 Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 23 Mar 2017, 06:54 Please correct me if i'm wrong. For the inequality |x-5| <= 2.5 to be +ve, x>5 For the inequality |x-5| <= 2.5 to be -ve, x<5 By removing the modulus, (x-5) = 2.5; x = 7.5 -(x-5) = 2.5; x = 2.5 Hence the inequality lies between 2.5 < (x-5) < 7.5 Since the question asks for integers within this range, we have 3,4,5,6 and 7. Hence answer is C. Is there any possibility this question be asked for non-integer values.? Just trying to understand the concept better. TIA _________________ Regards, Sarugiri Manager Joined: 02 Aug 2015 Posts: 80 Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 23 Mar 2017, 09:14 sarugiri wrote: Please correct me if i'm wrong. For the inequality |x-5| <= 2.5 to be +ve, x>5 For the inequality |x-5| <= 2.5 to be -ve, x<5 By removing the modulus, (x-5) = 2.5; x = 7.5 -(x-5) = 2.5; x = 2.5 Hence the inequality lies between 2.5 < (x-5) < 7.5 Since the question asks for integers within this range, we have 3,4,5,6 and 7. Hence answer is C. Is there any possibility this question be asked for non-integer values.? Just trying to understand the concept better. TIA Hello there, For the given set of options, the question cannot ask for non-integer values as there are infinite non integral values between any two integers. However, had the options were limits, then there's a possibility to ask for non-integeral values. Cheers! GMAT Forum Moderator Joined: 28 May 2014 Posts: 528 GMAT 1: 730 Q49 V41 How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 24 Mar 2017, 00:43 Diwakar003 wrote: Hello there, For the given set of options, the question cannot ask for non-integer values as there are infinite non integral values between any two integers. However, had the options were limits, then there's a possibility to ask for non-integeral values. Cheers! Hi, Could you please explain this in little more details. Cheers. _________________ Math Expert Joined: 02 Sep 2009 Posts: 46207 Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 24 Mar 2017, 01:03 saswata4s wrote: Diwakar003 wrote: Hello there, For the given set of options, the question cannot ask for non-integer values as there are infinite non integral values between any two integers. However, had the options were limits, then there's a possibility to ask for non-integeral values. Cheers! Hi, Could you please explain this in little more details. Cheers. The point is that there are infinitely many numbers in any interval. For example, if we don't limit the values of x to integers only, then infinitely many values of x satisfy 2.5 ≤ x ≤ 7.5. Does this make sense? _________________ Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2570 Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 27 Mar 2017, 11:54 saswata4s wrote: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 Let’s solve for when (x - 5) is positive and when (x - 5) is negative. When (x - 5) is positive: x - 5 ≤ 2.5 x ≤ 7.5 When (x - 5) is negative: -(x - 5) ≤ 2.5 -x + 5 ≤ 2.5 -x ≤ -2.5 x ≥ 2.5 Thus, 2.5 ≤ x ≤ 7.5. We see that there are 5 integer values that satisfy the inequality: 3, 4, 5, 6, and 7. Answer: C _________________ Jeffery Miller Head of GMAT Instruction GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 11803 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink] ### Show Tags 28 Mar 2017, 20:41 1 Hi saswata4s, A certain number of questions in the Quant section of the GMAT can be solved rather easily with 'brute force' arithmetic. You don't need to know/complete any complex math - you just have to put the pen on the pad and work your way through the possibilities. The answer choices to this question show that there are at least 3, but no more than 7, integer values that 'fit' this inequality. I bet that you can find them all in under a minute if you try. |X - 5| < 2.5 Let's start with an obvious value: X = 5 |5 - 5| = 0, which is clearly less than 2.5 X = 6 ---> |1| is less than 2.5 X = 7 ---> |7| is less than 2.5 X = 8 ---> |3| is NOT less than 2.5.... Increasing the value of X will just increase the value of the inequality, so there's no reason to check any higher integers. Let's stop here and try going in the other direction.... X = 4 ---> |-1| = 1 is less than 2.5 X = 3 ---> |-2| = 2 is less than 2.5 X = 2 ---> |-3| = 3 is NOT less than 2.5....Decreasing the value of X will just increase the value of the inequality, so there's no reason to check any lower integers. Thus, we're done - and there are 5 integer values that fit the inequality. Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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15 May 2017, 05:35
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

2.5<=x<=7.5

As x is integer so x can be 3,4,5,6,7

Hence 5 values can x take

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How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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10 Feb 2018, 07:05
Bunuel wrote:
saswata4s wrote:
How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

|x - 5| ≤ 2.5;

Get rid of modulus: -2.5 ≤ x - 5 ≤ 2.5;

Add 5 to all three parts: 2.5 ≤ x ≤ 7.5

Integer values of x that satisfy the inequality are 3, 4, 5, 6, and 7. So, 5 values.

Hello Bunuel, Zeus of all real numbers and unreal numbers please show your kindness to mortal GMAT student by explaining what i did wrong in my solution below:-)

I approached this question by opening the modulus - modulus always is associated with opening modulus no ?

|x - 5| ≤ 2.5

Case 1: x is positive x - 5≤ 2.5 --> x = 7.5

Case 2: x is negative x - 5≤ -2.5 --> x = 2.5 (Not valid, our initial condition was that x is negative but we got positive.

So whats wrong with my approach ?
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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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10 Feb 2018, 23:13
1
Hi dave13,

Your approach to solving this problem isn't "wrong", but it appears "incomplete."

You've determined the "upper limit" and lower limit" for what X could be (meaning that 2.5 <= X <= 7.5), but you did not factor in ALL of the given information. We're told that X MUST be an INTEGER, so given the range that you've determined... how many INTEGER values could X be?

X could be 3, 4, 5, 6 or 7... meaning that there are 5 potential INTEGER solutions to this question.

GMAT assassins aren't born, they're made,
Rich
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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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11 Feb 2018, 07:52
EMPOWERgmatRichC wrote:
Hi dave13,

Your approach to solving this problem isn't "wrong", but it appears "incomplete."

You've determined the "upper limit" and lower limit" for what X could be (meaning that 2.5 <= X <= 7.5), but you did not factor in ALL of the given information. We're told that X MUST be an INTEGER, so given the range that you've determined... how many INTEGER values could X be?

X could be 3, 4, 5, 6 or 7... meaning that there are 5 potential INTEGER solutions to this question.

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

Hello EMPOWERgmatRichC,

Many thanks for your reply :) you know what confused? :? CASE 2

Case 2: x is negative x - 5≤ -2.5 --> x = 2.5 (Not valid, our initial condition was that x is negative but we got positive

case two was not valid so that's why I thought I again did something wrong, case two was invalid...

could you explain the logic behind case TWO? what does "INVALID" mean ?

thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 46207
Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5? [#permalink]

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11 Feb 2018, 08:08
1
dave13 wrote:
Bunuel wrote:
saswata4s wrote:
How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

|x - 5| ≤ 2.5;

Get rid of modulus: -2.5 ≤ x - 5 ≤ 2.5;

Add 5 to all three parts: 2.5 ≤ x ≤ 7.5

Integer values of x that satisfy the inequality are 3, 4, 5, 6, and 7. So, 5 values.

Hello Bunuel, Zeus of all real numbers and unreal numbers please show your kindness to mortal GMAT student by explaining what i did wrong in my solution below:-)

I approached this question by opening the modulus - modulus always is associated with opening modulus no ?

|x - 5| ≤ 2.5

Case 1: x is positive x - 5≤ 2.5 --> x = 7.5

Case 2: x is negative x - 5≤ -2.5 --> x = 2.5 (Not valid, our initial condition was that x is negative but we got positive.

So whats wrong with my approach ?

When opening the modulus you should consider the EXPRESSION being positive or negative, not x. The simplest approach is given in my post but if we solve the way you are going then we'd have the following.

|x - 5| ≤ 2.5

When x - 5 < 0 (NOT x), so when x < 5, we'd have -(x - 5) ≤ 2.5 --> $$x \geq 2.5$$. Since we consider the range when x < 5, then for this range we'd have $$2.5 \leq x < 5$$

When x - 5 >= 0 (NOT x), so when x >= 5, we'd have x - 5 ≤ 2.5 --> $$x \leq 7.5$$. Since we consider the range when x >= 5, then for this range we'd have $$5 \leq x \leq 7.5$$

Combining both: $$2.5 \leq x \leq 7.5$$

You should study articles below carefully:

10. Absolute Value

[/list]

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?   [#permalink] 11 Feb 2018, 08:08
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