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# Solving Mods using diagrams

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Veritas Prep GMAT Instructor
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Solving Mods using diagrams [#permalink]  18 Apr 2012, 02:01
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Expert's post
Responding to a pm:

Question: What is the solution set for |3x-2|\leq|2x-5|?

Solution:

|3x-2| \leq |2x-5| is equivalent to 3|x-2/3| \leq 2|x-5/2|

Essentially, we want values of x for which thrice the distance from 2/3 is less than or equal to twice the distance from 5/2.

Put it on a number line

Attachment:

Ques3.jpg [ 12.35 KiB | Viewed 1054 times ]

Look at the first diagram. There will be a point between 2/3 (= 4/6) and 5/2 ( = 15/6) where thrice the distance from 2/3 is equal to twice the distance from 5/2. Basically, the distance between the 2 points (= 11/6) will be split in the ratio 2:3.
2/5 * 11/6 = 11/15
The point between them is 2/3 + 11/15 = 21/15 = 7/5
If we move toward left from 7/5, the distance shown by red lines will keep decreasing and that shown by green lines will keep increasing so all those points are acceptable. After you reach 4/6, the distance shown by red lines will start increasing till at a certain point, the distance covered by 3 red lines will be again equal to the distance covered by 2 green lines. Look at the second diagram.
The third red line will be equal to twice the distance of 11/6 i.e. this point will be 2*11/6 = 22/6 away from 4/6.

The point is 4/6 - 22/6 = -18/6 = -3 (figure not drawn to scale)
At -3, thrice the distance from 4/6 is equal to twice the distance from 5/2.
If you go further left from -3, the distance depicted by the red lines will start increasing so those points are not acceptable.

Therefore, -3 <= x <= 7/5

To verify, put in the points in the given inequality and check.
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Karishma
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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Manager Status: Do till 740 :) Joined: 13 Jun 2011 Posts: 102 Concentration: Strategy, General Management GMAT 1: 460 Q35 V20 GPA: 3.6 WE: Consulting (Computer Software) Followers: 0 Kudos [?]: 5 [0], given: 18 Re: Solving Mods using diagrams [#permalink] 18 Apr 2012, 09:39 Thanks a lot Karishma but I think this gets complex atleast at this stage for me to solve this using graphs . I shall practice more and get to this part. Thanks again Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4032 Location: Pune, India Followers: 860 Kudos [?]: 3618 [0], given: 144 Re: Solving Mods using diagrams [#permalink] 18 Apr 2012, 11:08 Expert's post shankar245 wrote: Thanks a lot Karishma but I think this gets complex atleast at this stage for me to solve this using graphs . I shall practice more and get to this part. Thanks again Conceptually, the question is easy but the numbers here make it unwieldy. I would suggest you to try the following questions (in this order): |x| = |x-4| |x| \leq |x-4| 2|x| = |x-6| 2|x| \leq |x-6| 2|x-1| \leq 3|x-11| It might make more sense to you after this. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Re: Solving Mods using diagrams [#permalink]  07 May 2012, 01:34
Hi Karishma,

Is there any other method to solve these kind of problems ?

Thx much,
Veritas Prep GMAT Instructor
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Location: Pune, India
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Re: Solving Mods using diagrams [#permalink]  07 May 2012, 02:23
Expert's post
tennisstar wrote:
Hi Karishma,

Is there any other method to solve these kind of problems ?

Thx much,

You can certainly use algebra to solve these questions. Divide the number line, check for positive negative regions and solve for various cases etc. I have given this approach because a member wanted the diagrammatic approach to this question (the diagrammatic approach is much faster and intuitive but only after you have a thorough understanding of what you are supposed to do and, more importantly, why). Search for mods in the forum and you will find many examples discussing algebraic approaches to such questions.
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Karishma
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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Senior Manager Joined: 07 Sep 2010 Posts: 341 Followers: 2 Kudos [?]: 48 [1] , given: 136 Re: Solving Mods using diagrams [#permalink] 07 May 2012, 07:08 1 This post received KUDOS Though you cant find better response than the Karishma's one, here is the algebraic approach - |3x-2|\leq|2x-5| Since it is inequality, you must remember that you cannot square both sides until and unless you are sure about the signs of both sides. We know that modulus is always positive, so both l.h.s and r.h.s can be squared and sign of inequality will remain same. This is what I remember when I try to solve such questions - It is okay to square both sides of the equation if you are sure that each side is greater than or equal to zero i.e |2x-5|^2 >= |3x-2|^2 Open the square and subtract one equation from another, you will get 5x^2+8x-21<=0 i.e 5(x-7/5)(x+3)<=0 ie. x belongs to (-3,7/5) let me know if somthing is not clear VeritasPrepKarishma wrote: Responding to a pm: Question: What is the solution set for |3x-2|\leq|2x-5|? _________________ +1 Kudos me, Help me unlocking GMAT Club Tests Senior Manager Joined: 07 Sep 2010 Posts: 341 Followers: 2 Kudos [?]: 48 [0], given: 136 Re: Solving Mods using diagrams [#permalink] 07 May 2012, 09:12 Hi Karishma, I tend to solve such questions using algebraic approach. Also, I know the basics of graphical approach and can apply well if the equation is follows pattern such as |x-a|+|x-b| = variable. However, for solving below questions, I usually fall flat. The solution that you have provided has given me some insights, but I failed to understand the below colored portion. Could you please explain that part a bit. i.e How did you infer this- The third red line will be equal to twice the distance of 11/6 i.e. this point will be 2*11/6 = 22/6 away from 4/6. Thanks H VeritasPrepKarishma wrote: Responding to a pm: Question: What is the solution set for |3x-2|\leq|2x-5|? Solution: |3x-2| \leq |2x-5| is equivalent to 3|x-2/3| \leq 2|x-5/2| Essentially, we want values of x for which thrice the distance from 2/3 is less than or equal to twice the distance from 5/2. Put it on a number line Attachment: Ques3.jpg Look at the first diagram. There will be a point between 2/3 (= 4/6) and 5/2 ( = 15/6) where thrice the distance from 2/3 is equal to twice the distance from 5/2. Basically, the distance between the 2 points (= 11/6) will be split in the ratio 2:3. 2/5 * 11/6 = 11/15 The point between them is 2/3 + 11/15 = 21/15 = 7/5 If we move toward left from 7/5, the distance shown by red lines will keep decreasing and that shown by green lines will keep increasing so all those points are acceptable. After you reach 4/6, the distance shown by red lines will start increasing till at a certain point, the distance covered by 3 red lines will be again equal to the distance covered by 2 green lines. Look at the second diagram. The third red line will be equal to twice the distance of 11/6 i.e. this point will be 2*11/6 = 22/6 away from 4/6. The point is 4/6 - 22/6 = -18/6 = -3 (figure not drawn to scale) At -3, thrice the distance from 4/6 is equal to twice the distance from 5/2. If you go further left from -3, the distance depicted by the red lines will start increasing so those points are not acceptable. Therefore, -3 <= x <= 7/5 To verify, put in the points in the given inequality and check. _________________ +1 Kudos me, Help me unlocking GMAT Club Tests Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4032 Location: Pune, India Followers: 860 Kudos [?]: 3618 [1] , given: 144 Re: Solving Mods using diagrams [#permalink] 08 May 2012, 08:17 1 This post received KUDOS Expert's post imhimanshu wrote: Hi Karishma, I tend to solve such questions using algebraic approach. Also, I know the basics of graphical approach and can apply well if the equation is follows pattern such as |x-a|+|x-b| = variable. However, for solving below questions, I usually fall flat. The solution that you have provided has given me some insights, but I failed to understand the below colored portion. Could you please explain that part a bit. i.e How did you infer this- The third red line will be equal to twice the distance of 11/6 i.e. this point will be 2*11/6 = 22/6 away from 4/6. Thanks H Look at the second diagram very carefully. We want that the sum of the lengths of the red lines to be equal to the sum of the lengths of the green lines. Note that the red lines are shorter than the green lines. Then how will the sum of the length of the red lines be equal to the sum of the length of the green lines? Thankfully, we have an extra red line (3 red lines vs 2 green lines). This extra red line will need to make up for the extra length of the green lines as compared to the length of the red lines. This one extra red line needs to cover the extra length of both the green lines. What is the extra length of both the green lines together? It is 2* (11/6) (look at the diagram). Hence, the length of the red line = 22/6 So the leftmost point must be at a distance 22/6 to the left of 4/6. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Re: Solving Mods using diagrams [#permalink]  08 May 2012, 09:30
Awesum Post. Kudos. Thanks a lot for the explanation.

VeritasPrepKarishma wrote:
imhimanshu wrote:
Hi Karishma,
I tend to solve such questions using algebraic approach. Also, I know the basics of graphical approach and can apply well if the equation is follows pattern such as |x-a|+|x-b| = variable.
However, for solving below questions, I usually fall flat. The solution that you have provided has given me some insights, but I failed to understand the below colored portion.
Could you please explain that part a bit.
i.e How did you infer this-

The third red line will be equal to twice the distance of 11/6 i.e. this point will be 2*11/6 = 22/6 away from 4/6.

Thanks
H

Look at the second diagram very carefully. We want that the sum of the lengths of the red lines to be equal to the sum of the lengths of the green lines. Note that the red lines are shorter than the green lines. Then how will the sum of the length of the red lines be equal to the sum of the length of the green lines? Thankfully, we have an extra red line (3 red lines vs 2 green lines). This extra red line will need to make up for the extra length of the green lines as compared to the length of the red lines. This one extra red line needs to cover the extra length of both the green lines. What is the extra length of both the green lines together? It is 2* (11/6) (look at the diagram). Hence, the length of the red line = 22/6
So the leftmost point must be at a distance 22/6 to the left of 4/6.

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Re: Solving Mods using diagrams   [#permalink] 08 May 2012, 09:30
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