Modulus Question

Expert replies

Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Thu Oct 20, 2016 6:21 pm

What is the value of x?

(1) |6 - 3x| = x - 2

(2) |5x + 3| = 2x + 9

OA: A

Source: Magoosh

Is it the only solution is by using the rule: if |a|= b then a=b or -a=b. Is other way to solve this question?

Like this post

Source: Beat The GMAT — Data Sufficiency | Thu Oct 20, 2016 6:21 pm

GMAT/MBA Expert

ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Thu Oct 20, 2016 7:24 pm

Absolute value questions can be tricky. We know that the rule (stating it a little bit differently than you did, but same idea) is that if |a| = b, then a = b or a = -b. So, if we had a DS question that asked "what is the value of x?", then a statement that said "|x| = 3" would not be sufficient, because x could be 3 or -3.

It's tempting to think that any time you're presented with an absolute value DS question, therefore, a single statement would have to be insufficient. However, there are a few exceptions...

1. An absolute value equation might have only one solution (rather than two) if one of the solutions, when plugged back into the equation, doesn't work. This will only happen when you have the same variable both inside and outside the absolute value sign. You can't assume - you have to test them.

2. When you solve both the positive and negative case, it's possible to end up with 2 solutions that are in fact the same value. This is similar to solving quadratics: usually we get 2 solutions, but with perfect squares we get 2 of the same solution.

3. If your absolute value is set equal to zero, as in |y - 4| = 0. You can't have "positive 0" and "negative 0," so you'll just get a single solution.

(1) |6 - 3x| = x - 2
When we solve, we get 2 solutions:

positive case:
6 - 3x = x - 2
8 = 4x
x = 2
Plug it back in to test:
|6 - 3(2)| = 2 - 2
|0| = 0 --> this works!

negative case:
6 - 3x = -(x - 2)
6 - 3x = -x + 2
4 = 2x
x = 2

Our 2 solutions were the same value, so this statement is actually sufficient. (And we don't need to test again for statement 2).

(2) |5x + 3| = 2x + 9

positive case:
5x + 3 = 2x + 9
3x = 6
x = 2
Plug back in:
|5(2) + 3| = 2(2) + 9
|13| = 13 --> this works

negative case:
5x + 3 = -(2x + 9)
5x + 3 = -2x - 9
7x = -12
x = -12/7
Plug back in:
|5(-12/7) + 3| = 2(-12/7) + 9
|-60/7 + 21/7| = -24/7 + 63/7
|-39/7| = 39/7 --> this works!

Because we got 2 different solutions that worked, this statement is insufficient.

The answer is A.

Did this answer your question?

Last edited by ceilidh.erickson on Fri Oct 21, 2016 4:23 pm, edited 1 time in total.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

Like this post

GMAT/MBA Expert

ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Thu Oct 20, 2016 7:27 pm

Here's an example of exception #1 that I listed above:
https://www.beatthegmat.com/simple-yet-a ... tml#714086

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

Like this post

crackverbal: Master | Next Rank: 500 Posts; Posts: 157; Joined: Mon Aug 16, 2010 7:30 pmFollowed by: 3 members; Location: India; Thanked: 65 times

by crackverbal » Thu Oct 20, 2016 9:25 pm

Hi Mo2men,

Let me try to explain these types of questions in GMAT little more elaborately.

There are basically you could expect four types of modulus questions (particularly in DS).

1. LHS is a modulus of some equation and the RHS is just a equation and the question is a YES-NO DS question
2. LHS is modulus of some equation and the RHS is just a equation and the question is a Value of DS question
3. Modulus and inequalities.
4. Both side modulus.

This question is of the second type here. I will explain at the last how we can solve when asked of the first type.

General rule of a modulus (absolute value),

|x| = x , whenever x > = 0

|x| = (-x), whenever x < 0

Here in this question,

We have to find the value of x,

I. |6-3x| = x-2

First case,

6-3x = x-2 , whenever (6-3x) > =0, that is x < = 2

4x = 8,

So x= 2 which matches condition x < = 2

Second case,

6-3x = -(x-2) , whenever (6-3x) < 0 , that is x > 2

Again this gives the same value.

So there is only value possible is 2.

So statement I is sufficient.

Eliminate the answer choices B, C and E. Keep the answer choices A and D.

II. |5x + 3| = 2x + 9

First case,

5x + 3 = 2x + 9 , whenever 5x >=3 , that is x >= (3/5)

3x = 6

x = 2 which matches the condition x >= (3/5)

Second case,

5x + 3 = -( 2x + 9) , whenever 5x <3 , that is x < (3/5)

7x = -12

x = -12/7, which matches the condition x < (3/5)

So we get here 2 values of "x".

So not sufficient.

So the answer is A.

Now, if this question of the first type which I mentioned above, lets see how to do it.

That is YES-NO DS question.

Let say the question is like this,

Is x > 0 ?

I. |6 - 3x| = x - 2

II. |5x + 3| = 2x + 9

Now for this question, we no need to do all the process we need above,

We know that LHS in the |6 - 3x| = x - 2 cannot be negative.

So we write,

x -2 > = 0

that says x >= 2

which is positive. sufficient.

Now the second statement,

|5x + 3| = 2x + 9, again LHS cannot be negative.

So 2x+9 >= 0,

So , x > = -9/2

So it could be negative or positive or zero.

So not sufficient.

So NET-NET, it depends on what question type of modulus GMAT asks. So you could smartly reduce time you spend
on questions when you know how to do the above types clearly.

Hope this is clear.

Join Free 4 part MBA Through GMAT Video Training Series here -
https://gmat.crackverbal.com/mba-throug ... video-2018

Enroll for our GMAT Trial Course here -
https://gmatonline.crackverbal.com/

For more info on GMAT and MBA, follow us on @AskCrackVerbal

Like this post

Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Fri Oct 21, 2016 4:18 am

Dear Ceilidh,

Thanks for your explanation.

Dear Crackverbal,

You exactly hit the point I wanted to discuss. I thought that we can solve type I using the same way of type II. I solved my question above as follows:

What is the value of X?

I.|6-3x| = x-2

We know that LHS in the |6 - 3x| = x - 2 cannot be negative.

So we write,

x -2 > = 0

that says x >= 2

So we can conclude that we have ONE solution. So Sufficient

II.|5x + 3| = 2x + 9

So 2x+9 >= 0,

So , x > = -9/2

So it could be negative or positive or zero.

So we can conclude that we have two solutions solution. so Insufficient

What do you think?

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Oct 21, 2016 4:23 am

|6x-3|

We need to consider two cases if the expression inside the absolute value can represent both a positive and a negative value.
Case 1: If 6-3x â‰¥ 0, then |6x-3| = 6-3x.
Case 2: If 6-3x < 0, then |6x-3| = -6+3x.

An example of Case 1: x=0
Here, 6-3x â‰¥ 0.
As a result, |6-3x| = 6-3x:
|6 - 3*0| = 6 - 3*0
6 = 6.

An example of Case 2: x=3
Here, 6-3x < 0.
As a result, |6-3x| = -6+3x:
|6 - 3*3| = -6 + 3*3
3 = 3.

Statement 1: |6 - 3x| = x - 2

In this equation, since the absolute value on the left side cannot be equal to a negative value, the right side must be NONNEGATIVE:
x-2â‰¥0, implying that xâ‰¥2.

Given this constraint, only Case 2 is possible.
Implication:
|6 - 3x| = x - 2 has only ONE solution.
Since the value of this solution can be determined, Statement 1 is SUFFICIENT.

Solution:
In Case 2, |6-3x| = -6+3x.
Substituting |6-3x| = -6+3x into |6-3x| = x-2, we get:
-6+3x = x-2
2x = 4
x = 2.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review