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[|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s

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nick1816
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[|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   22 May 2021, 18:19
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\([ |\frac{2x-3}{4}| ] = 5\)

, where [x] denotes the greatest integer that is smaller than or equal to x. Find the number of integers that satisfy the above equation.

A. 0
B. 1
C. 2
D. 3
E. 4
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RamseyGooner
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   23 May 2021, 13:36
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I have always had trouble with Modulus and inequalities, here goes my translation -

[x] denotes the greatest integer that is smaller than or equal to x

[6.9] = 6, [5.5] = 5, [5]= 5 etc

5 <= |(2x-3)/4| < 6
20 <= |2x-3| < 24

Now taking mod as +,
23 <= 2x < 27; x can be 12 or 13

Now taking mod as -,
17 <= -2x < 21; x can be -9 or -10

So , x can be -9,-10, 12, 13

Hope this helps!
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ntngocanh19
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[|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   Updated on: 23 May 2021, 08:59
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nick1816 wrote:
\([ |\frac{2x-3}{4}| ] = 5\)

=>4 < |(2x-3)/4| <=5
=> 16 < |2x-3|<= 20
So , x can be -7,-8, 10, 11
=> hence choice E

Posted from my mobile device

Originally posted by ntngocanh19 on 22 May 2021, 18:42.
Last edited by ntngocanh19 on 23 May 2021, 08:59, edited 1 time in total.
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   23 May 2021, 08:47
ntngocanh19 wrote:
nick1816 wrote:
\([ |\frac{2x-3}{4}| ] = 5\)

=>4 < |(2x-3)/4| <=5
=> 16 < |2x-3||<= 20
So , x can be -7,-8, 10, 11
=> hence choice E

Posted from my mobile device


Hello, could I get your help to share why is >4 the lower limit?

Thank you!
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ntngocanh19
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   23 May 2021, 08:55
joyousjoyce wrote:
ntngocanh19 wrote:
nick1816 wrote:
\([ |\frac{2x-3}{4}| ] = 5\)

=>4 < |(2x-3)/4| <=5
=> 16 < |2x-3||<= 20
So , x can be -7,-8, 10, 11
=> hence choice E

Posted from my mobile device


Hello, could I get your help to share why is >4 the lower limit?

Thank you!

It's because the definition of [x] is the greatest integer that is smaller than or equal to x.
Here we have [A] = 5 (A=|(2x-3)/4|)
Then if A<=4 <=> [A] =4
=> A must be greater than 4

Hope this help! :D
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   23 May 2021, 09:38
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ntngocanh19 wrote:
joyousjoyce wrote:
nick1816 wrote:
\([ |\frac{2x-3}{4}| ] = 5\)


Hello, could I get your help to share why is >4 the lower limit?

Thank you!

It's because the definition of [x] is the greatest integer that is smaller than or equal to x.
Here we have [A] = 5 (A=|(2x-3)/4|)
Then if A<=4 <=> [A] =4
=> A must be greater than 4

Hope this help! :D


Thanks for helping to explain in more details. I think I am having challenges understanding the limit... let me try to share my thinking.

[x] is the greatest integer that is smaller than or equal to x
x = 2.5, [x] = 2

In the question above [A] = 5, so shouldn't it be that A = 5~5.9?
[5.5] = 5
[5.9] = 5

For A<=4, [4] = 4, and [4.5] = 4. Hence A has to be at least 5.

Could you share with me the flaw in my thinking? Thank you..
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ntngocanh19
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   23 May 2021, 09:51
joyousjoyce wrote:

Thanks for helping to explain in more details. I think I am having challenges understanding the limit... let me try to share my thinking.

[x] is the greatest integer that is smaller than or equal to x
x = 2.5, [x] = 2

In the question above [A] = 5, so shouldn't it be that A = 5~5.9?
[5.5] = 5
[5.9] = 5

For A<=4, [4] = 4, and [4.5] = 4. Hence A has to be at least 5.

Could you share with me the flaw in my thinking? Thank you..

I think you are right. A
I found out that I'm confused between [x] and x.
Thank you for pointing out :D
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Sevil92
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink] New post   01 Nov 2022, 07:57
RamseyGooner wrote:
I have always had trouble with Modulus and inequalities, here goes my translation -

[x] denotes the greatest integer that is smaller than or equal to x

[6.9] = 6, [5.5] = 5, [5]= 5 etc

5 <= |(2x-3)/4| < 6
20 <= |2x-3| < 24

Now taking mod as +,
23 <= 2x < 27; x can be 12 or 13

Now taking mod as -,
17 <= -2x < 21; x can be -9 or -10

So , x can be -9,-10, 12, 13

Hope this helps!


Hello everyone!

Unfortunately, explanations did not help me get the logic here. I just solved the equation in traditional manner (left side with positive x equals to 5; left side with negative x equals to 5) and got [x=11.5] and [-8.5] respectively, and ended up that x equals 11 and -9 (smaller integers than x itself). Can anyone correct me? I cannot see why we take 4 as a limit.
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Re: [|2x-3/4|] = 5[/m] , where [x] denotes the greatest integer that is s [#permalink]
01 Nov 2022, 07:57
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