For how many values of x, is |||x - 5| -10| -5| = 2? : Problem Solving (PS)

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KarishmaB

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Re: Question of the Day [#permalink]

04 Jul 2012, 00:20

Expert Reply

Eshaninan wrote:

Not using graphs:
Take |x-5| as "A" , |A-10| as "B"
then |B-5| = 2 would have 2 answers 7 and 3 for B
Now, take |A-10|=7, would give two solutions for A and |A-10|=3 would give 2 solutions for A
For each of these 4 solutions equate for |x-5| which would give 2 solutions for each equation and hence a total of 4*2 = 8 solution for x

Yes, you can do it this way too. But you have to be careful of the values given e.g.
In instead of 2, we have 0, you will get 4 solutions since |B-5| = 0 will give you only 1 value of B. Each of the subsequent values will give you 2 values each. I am not a big fan of multiple variables so I don't use them. Graphical approach just seems more intuitive to me.
Anyway, as long as you know what you are doing, you can use whatever method you like.
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Re: Question of the Day [#permalink]

08 Jul 2012, 07:20

anujkch wrote:

For equation mentioned by Gurpreet i.e. |||x-5|-10|-5| = 6 can you please explain, without graph, how you were able to get answer as 2.

Following one of the post with mathematical values

|||x-5|-10|-5| =6
let |x-5| = a which makes above
||a-10|-5| =6

let |a-10| = b which makes
|b-5| = 6

now for the above b can take 1, 11
for every b =1 a can have 9, 11
and for b = 11 a can have 1 and 21

so 'a' has four solutions 1, 9, 11 and 21

Thus for a as 1 x can take 4,6
Thus for a as 9 x can take 4 , 14
Thus for a as 11 x can take 6 , 14
Thus for a as 21 x can take 16 , 24

Total - 5 different values.

First you need to understand what is the graph. you may check karishma's solution for that.

Now when you have graph ready you need to see where these two graphs intersect.
lllx - 5l -10l -5l = 0 and Y=2

since the max value of Y = 5 in between x = -10,0,10,20 => Y=2 will touch 8 times.
but when Y=6 it would touch only 2 times. the extreme ones after x=20 and before x= -10

Just look at the attachment of karishma.

PS: This question is very good. I would really love to see Gmat testing on such questions.

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bagdbmba

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Re: Question of the Day [#permalink]

01 Dec 2012, 08:45

VeritasPrepKarishma wrote:

Question of the Day:

Q. For how many values of x, is lllx - 5l -10l -5l = 2?
(A) 0
(B) 2
(C) 4
(D) 8
(E) More than 8

(Those ls are mods)
(Earn kudos if you solve it in under 2 mins.)
Note: This question is beyond GMAT level. Nevertheless, it helps you understand mods and graphs and hence, is great for practice.)

Hi Karishma,
Your sig. in this post contains a link that says "Download the Veritas Prep GMAT On Demand App Free https://itunes.apple.com/us/app/gmat-on- ... 60224?mt=8".

But I think veritas Prep on Demand course is NOT available FREE to download on apple product.

Your thoughts on this..!!

mbaiseasy

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Re: Question of the Day [#permalink]

06 Dec 2012, 05:33

Crazy question haha! But I had fun!

|||x-5|-10|-5| = 2

Let y = ||x-5|-10|
|y-5| = 2 gives us two values of y=7 and y=3

Let w = |x-5|
|w - 10|=7 or |w-10|=3
Each will give two values of w. Thus, we have 4 values of w: 17,3,13,7

|x-5|=17, x=22 and -12
|x-5|=3, x=8 and 2
|x-5|=13, x=18 and -8
|x-5|=7, x=12 and -2

Answer: 8

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Re: Question of the Day [#permalink]

06 Dec 2012, 06:24

shortcut:

there are tow possible x for |x-z|=y as long as y=|=0 since the result can be either positive or negative.

therefore we have
2x2x2=8
possible solutions

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KarishmaB

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Re: Question of the Day [#permalink]

06 Dec 2012, 21:55

Expert Reply

debayan222 wrote:

Hi Karishma,
Your sig. in this post contains a link that says "Download the Veritas Prep GMAT On Demand App Free https://itunes.apple.com/us/app/gmat-on- ... 60224?mt=8".

But I think veritas Prep on Demand course is NOT available FREE to download on apple product.

Your thoughts on this..!!

The app is free. You can download it. The first lesson which discusses Test structure, scoring, pacing, myths, Veritas methodologies, typical Veritas class etc is free. Thereafter, you can choose the modules you want to work on. You will obviously need to pay for them. You can buy the modules you want instead of buying the whole course.
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Re: For how many values of x, is |||x - 5| -10| -5| = 2 is |||x? [#permalink]

14 Feb 2014, 22:57

Hi , from my thinking i got 3 x values is |||x - 5| -10| -5| = 2 ,
Uisng x = 8 , 2 , -12
Steps 1 |x - 5| = | 8 - 5 | = 3
step 2 | 3- 10 | = | -7 | = 7
step 3 | 7 -5 | = | 2 | = 2 :-D

in same way with x values 2 , and -12
karishma can you pls see my approach and reply me back
but u myself i think iam wrong

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KarishmaB

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Re: For how many values of x, is |||x - 5| -10| -5| = 2 is |||x? [#permalink]

16 Feb 2014, 21:07

Expert Reply

kanusha wrote:

Hi , from my thinking i got 3 x values is |||x - 5| -10| -5| = 2 ,
Uisng x = 8 , 2 , -12
Steps 1 |x - 5| = | 8 - 5 | = 3
step 2 | 3- 10 | = | -7 | = 7
step 3 | 7 -5 | = | 2 | = 2 :-D

in same way with x values 2 , and -12
karishma can you pls see my approach and reply me back
but u myself i think iam wrong

How did you get the values 8, 2 and -12? Note that there are a total of 8 values that x can take. Check the solutions given on the previous page and the attachment I have provided with the graphical approach.
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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

25 Apr 2015, 19:34

solved but in 2:52 mins.. what is the least time consuming way to approach this question?

ggurface

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

11 Aug 2015, 15:56

Here is an alternative to the graphical method already posted:

Start from the outside:
|a-5| = 2, where a = ||x-5| - 10| => a = 7,3
|b-10| = 7,3 where b = |x-5| => b = 17, 3, 13, 7
finally
|x-5| = 17,3,13,7. at this point, you do not have to solve it. we have the pattern where we first had 2 solutions for the outer ||, 4 for for the inner ||, we can assume 8 is the solution.

if 7 was a valid answer and you used this method, you need to continue it one further step to make sure that the solution wasnt duplicated.

mathivanan

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

13 Aug 2015, 06:24

1

Kudos

VeritasPrepKarishma wrote:

Question of the Day:

For how many values of x, is |||x - 5| -10| -5| = 2?

(A) 0
(B) 2
(C) 4
(D) 8
(E) More than 8

(Those ls are mods)
(Earn kudos if you solve it in under 2 mins.)
Note: This question is beyond GMAT level. Nevertheless, it helps you understand mods and graphs and hence, is great for practice.)

|x-5| has two values, therefore 2 solutions
There are 3 mod
Hence 2^3 = 8 solutions

dreamrunner

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

22 Aug 2015, 05:06

shrouded1 wrote:

Easiest way is to graph it, which will immediately show exactly 8 solutions. Alternatively :
|x-5|=0 .. Exactly 1 solution at x=5
||x-5|-10|=0 .. Exactly 2 solutions at x=5-10=-5 & x=5+10=15
|||x-5|-10|-5|=0 .. Exactly 4 solutions at x=-5-5,-5+5,15-5,15+5 OR x=-10,0,10,20

At this point it is very easy to imagine what the graph looks like, it'll be 3 triangular humps touching the x-axis at the above points and going to +inf as x goes below -10 or above 20.
All the lines are at 45 degrees, so the peaks will be at y=5

Now |||x-5|-10|-5|=2 is just equivalent to shifting the x-axis up by 2 units, or counting the number of intersections with the y=2 line. Given the peaks are at y=5, there will be the maximum possible number of intersections, which is 8 in the following regions :

Below -10
Between -10 & -5
Between -5 and 0
Between 0 & 5
Between 5 & 10
Betweem 10 & 15
Between 15 & 20
Above 20

Answer is 8.

As already mentioned, if you just graph it out, this should be straight forward to see

Can you please draw a graph for this question. very weak in this topic

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Bunuel

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Posts: 92698

Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

22 Aug 2015, 05:12

1

Kudos

Expert Reply

dreamrunner wrote:

shrouded1 wrote:

Easiest way is to graph it, which will immediately show exactly 8 solutions. Alternatively :
|x-5|=0 .. Exactly 1 solution at x=5
||x-5|-10|=0 .. Exactly 2 solutions at x=5-10=-5 & x=5+10=15
|||x-5|-10|-5|=0 .. Exactly 4 solutions at x=-5-5,-5+5,15-5,15+5 OR x=-10,0,10,20

At this point it is very easy to imagine what the graph looks like, it'll be 3 triangular humps touching the x-axis at the above points and going to +inf as x goes below -10 or above 20.
All the lines are at 45 degrees, so the peaks will be at y=5

Now |||x-5|-10|-5|=2 is just equivalent to shifting the x-axis up by 2 units, or counting the number of intersections with the y=2 line. Given the peaks are at y=5, there will be the maximum possible number of intersections, which is 8 in the following regions :

Below -10
Between -10 & -5
Between -5 and 0
Between 0 & 5
Between 5 & 10
Betweem 10 & 15
Between 15 & 20
Above 20

Answer is 8.

As already mentioned, if you just graph it out, this should be straight forward to see

Can you please draw a graph for this question. very weak in this topic

Here is a graph of |||x - 5| -10| -5| = 2:

Attachment:

MSP55621c3692c156b65ec00004239b89491b925i9.gif [ 5.74 KiB | Viewed 2918 times ]

Theory on Abolute Values: math-absolute-value-modulus-86462.html
Absolute value tips: absolute-value-tips-and-hints-175002.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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KarishmaB

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

24 Aug 2015, 00:01

Expert Reply

dreamrunner wrote:

shrouded1 wrote:

Easiest way is to graph it, which will immediately show exactly 8 solutions. Alternatively :
|x-5|=0 .. Exactly 1 solution at x=5
||x-5|-10|=0 .. Exactly 2 solutions at x=5-10=-5 & x=5+10=15
|||x-5|-10|-5|=0 .. Exactly 4 solutions at x=-5-5,-5+5,15-5,15+5 OR x=-10,0,10,20

At this point it is very easy to imagine what the graph looks like, it'll be 3 triangular humps touching the x-axis at the above points and going to +inf as x goes below -10 or above 20.
All the lines are at 45 degrees, so the peaks will be at y=5

Now |||x-5|-10|-5|=2 is just equivalent to shifting the x-axis up by 2 units, or counting the number of intersections with the y=2 line. Given the peaks are at y=5, there will be the maximum possible number of intersections, which is 8 in the following regions :

Below -10
Between -10 & -5
Between -5 and 0
Between 0 & 5
Between 5 & 10
Betweem 10 & 15
Between 15 & 20
Above 20

Answer is 8.

As already mentioned, if you just graph it out, this should be straight forward to see

Can you please draw a graph for this question. very weak in this topic

Another graphical solution is given here: for-how-many-values-of-x-is-x-103766.html#p808780

However, note that if you are weak in this topic, you may not want to start out here. Check out the more basic discussions on this topic first. This question is beyond GMAT level.
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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

10 Nov 2015, 16:17

Hi Bunuel.
Actually I still can not understand the right framework for this kind of questions ( how many solutions of X - ranges in modulus and inequalities. when I make the expression -ve and when +ve in the multi expression questions. please help in this and try to focus on the basics and the main concept to be able to answer such kind of questions.
many thanks in advance.

emonmisra

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

19 May 2019, 00:07

IIIx-5I-10I-5I= 2

consider IIx-5I-10I = t
It-5I=2 , t can have 2 value 7 or 3
Considering t has value 7, Ix-5I = a, |a-10I = 7 , so a can be either 17 or3,
Considering, 2nd option a can be 7,13
for each 4 values of a there can be 2 sets of value of x

Hence total 8 possible results.

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For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

09 Nov 2019, 00:48

$||x-5|-10|=3$ Or $||x-5|-10|=7$

Therefore

$|x-5|=13$ Or $|x-5|=7$

Therefore

$x=18, x=-8, x=12, x=-2$

VeritasKarishma

What is wrong here?
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swatib28

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

14 May 2021, 00:51

Are values of x = {- 17, -7,13,-3,8,-18,2, 12 } ?? Can anyone please confirm.. Thanks in advance.

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Re: For how many values of x, is |||x - 5| -10| -5| = 2? [#permalink]

23 Jun 2022, 19:50

Asked: For how many values of x, is |||x - 5| -10| -5| = 2?

Case 1: ||x - 5| -10| -5 = 2
||x - 5| -10| = 7
Case 1a : |x - 5| -10 = 7
|x - 5| = 17
x = 22 or - 12
Case 1b: |x - 5| -10 = -7
|x-5| = 3
x = 8 or 2
Case 2: ||x - 5| -10| -5 = -2
||x - 5| -10| = 3
Case2a: |x - 5| -10 = 3
|x -5| = 13
x = 18 or -8
Case 2b: |x - 5| -10 = -3
|x-5| = 7
x = 12 or -2

x = {22, -12, 8,2, 18,-8,12, -2}

IMO D

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