Inequalities - is the way im solving correct?

[onesome]

Hi - Can someone please help in understanding the strategy to be used to solve the inequalities question mentioned below. I think i did reach the correct answer but i'm not sure if i am solving it correctly.

Question =>
If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?
A) 3
B) 4
C) 5
D) 6
E) 7

How did i solve:
Step 1: Assuming the modulus will result in +ve value :
3 < x-5 <7
8 < x < 12
Possible values = 9,10,11 , i.e. 3 different values..........(A)

Step 2: Assuming the modulus will result in -ve value :
3 < -(x-5) < 7
3 < -x+5 < 7
solving , 3-5 < -x+5-5 < 7-5
to be finally , -2 < x < 2
Possible values = -1,0,1 , i.e. 3 different values..........(B)

From (A) and (B)
Total 6 different possible values,
Answer = D

Kindly suggest if you think something is wrong above.
Thanks in advance.

[rs2010]

That is the correct way.

[Ian Stewart]

That is the correct way.

I'd prefer to say "That is a correct way"!

The above solution is certainly correct, but as with most math problems, there is more than one solution. If you understand that |a - b| measures the distance between a and b on the number line, you can look at the problem as follows:

3 < |x-5| < 7

This is reallly two inequalities:

3 < |x-5|, so x is more than 3 away from 5, on the left or the right: x < 2 or x > 8.

|x-5| <7, so x is less than 7 away from 5, on the left or the right: x > -2, or x < 12.

Combining these, we have -2 < x < 2 or 8 < x < 12. If x is an integer, x can be -1, 0, 1, 9, 10 or 11. Drawing the number line makes this easier, of course.

[rs2010]

Thanks Ian. I will remember it.

[ildude02]

Ian,

Why do we have to consider the negative value for |x -5| ? Since it's given that | x- 5| is greater than 3 and and less then 7, doesn't it mean it lies with a postive range? So, (x-5) can only be postive, implying (x-5) > 0. So, why to consider negative values for |x-5| as in x-5 < 0? As a rule of thumb, do we always need to consider -ve value as well for a variable within the MOD.. I know you use distance relation to sovle MOD's, but it would be too much for me to digest this late But I still hope you could answer this, I would really appreciate your response.

That is the correct way.

I'd prefer to say "That is a correct way"!

The above solution is certainly correct, but as with most math problems, there is more than one solution. If you understand that |a - b| measures the distance between a and b on the number line, you can look at the problem as follows:

3 < |x-5| < 7

This is reallly two inequalities:

3 < |x-5|, so x is more than 3 away from 5, on the left or the right: x < 2 or x > 8.

|x-5| <7, so x is less than 7 away from 5, on the left or the right: x > -2, or x < 12.

Combining these, we have -2 < x < 2 or 8 < x < 12. If x is an integer, x can be -1, 0, 1, 9, 10 or 11. Drawing the number line makes this easier, of course.

[Stuart@KaplanGMAT]

My "too much math" alarm is sounding! I've always been a fan of the quick and dirty solution to GMAT questions.

If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?

All we need to do is count the number of values of x. Since we don't care what the values are, we can answer this question with minimal math.

Once we see that the "-5" is just going to shift our place on the number line (and since only integers are involved, we don't have to worry about anything wacky happening), we realize that it's irrelevant to answering the question.

Let's rephrase the question as:

If 3 < |x| < 7 , where x is an integer, how many possible values does x have?

How many integers are there between 3 and 7? 4, 5 and 6, so "three".

Because it's absolute value, we know there will be two solution sets. So, -4, -5 and -6 will also work. That's a total of 6: choose (d).

[ildude02]

Thanks for the response. I do beleive "too much math" is kinda hampering even my basic skills

I realized after I posted the question that I got confused between regular (X-5)vs |X-5|, since X-5 has to be postive if X is an integer if there wasn't a MOD. But it's good that I realized I could even make mistakes like this.

[Ian Stewart]

My "too much math" alarm is sounding! I've always been a fan of the quick and dirty solution to GMAT questions.

If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?

That's all fine for this question, but what if you're asked:

What is the area of the region in the (x,y)-plane consisting of all points for which 3 <= |x-5| <= 7 and |y| <= 2 ?

Or:

What is the longest distance between two points which both satisfy:
3 <= |x-5| <= 7 and |y| <= 2

I think there's a lot of value in learning how to work out the values of x which satisfy the absolute value equation, at least if you're aiming for a top Quant score.

What I'm really saying is that if you're aiming for a 90th+ percentile in GMAT math, turn your 'too much math' alarm off- you're going to need to do some math. It's when the math would take longer than two minutes that you should look for another approach. I don't see at all how this question could take close to that long, so it's certainly not a question of 'too much math'. And if you want to do well on difficult absolute value questions, understand what absolute value measures: distance. Every high-level GMAT absolute value question I've seen is testing exactly that, and if you understand that well, these problems are not difficult.

[wawatan]

by the way if you don't know if you did the second statement correct,you can plug values back into x to see if it satisfies the equation .

to be finally , -2 < x < 2
Possible values = -1,0,1 , i.e. 3 different values

3 < |x-5| < 7 so put 0=x, you notice how the equation will become
3 < 5 < 7...so you know you solved it correctly.

[ildude02]

Ian, thanks for your response. I didn't mean this question was "too much" of math. I have been juggling around too many concepts and doing lot of math lately, so that's what I was referring to. I'm trying to stick with the concepts that I'm strong at, but at the same time, I'm learning new concepts if I feel I'm weak in a particular area. So all your reponses are appreciated and I only wish I had more time before my exam, since I learn some new technique's every other day on this forum.

My "too much math" alarm is sounding! I've always been a fan of the quick and dirty solution to GMAT questions.

If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?

That's all fine for this question, but what if you're asked:

What is the area of the region in the (x,y)-plane consisting of all points for which 3 <= |x-5| <= 7 and |y| <= 2 ?

Or:

What is the longest distance between two points which both satisfy:
3 <= |x-5| <= 7 and |y| <= 2

I think there's a lot of value in learning how to work out the values of x which satisfy the absolute value equation, at least if you're aiming for a top Quant score.

What I'm really saying is that if you're aiming for a 90th+ percentile in GMAT math, turn your 'too much math' alarm off- you're going to need to do some math. It's when the math would take longer than two minutes that you should look for another approach. I don't see at all how this question could take close to that long, so it's certainly not a question of 'too much math'. And if you want to do well on difficult absolute value questions, understand what absolute value measures: distance. Every high-level GMAT absolute value question I've seen is testing exactly that, and if you understand that well, these problems are not difficult.

[ildude02]

Ian, thanks for your response. I didn't mean this question was "too much" of math. I have been juggling around with too many concepts and doing lot of math lately, so that's what I was referring to. I'm trying to stick with the concepts that I'm strong at, but at the same time, I'm learning new concepts if I feel I'm weak in a particular area. So all your reponses are appreciated and I only wish I had more time before my exam, since I learn some new technique's every other day on this forum.

My "too much math" alarm is sounding! I've always been a fan of the quick and dirty solution to GMAT questions.

If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?

That's all fine for this question, but what if you're asked:

What is the area of the region in the (x,y)-plane consisting of all points for which 3 <= |x-5| <= 7 and |y| <= 2 ?

Or:

What is the longest distance between two points which both satisfy:
3 <= |x-5| <= 7 and |y| <= 2

I think there's a lot of value in learning how to work out the values of x which satisfy the absolute value equation, at least if you're aiming for a top Quant score.

What I'm really saying is that if you're aiming for a 90th+ percentile in GMAT math, turn your 'too much math' alarm off- you're going to need to do some math. It's when the math would take longer than two minutes that you should look for another approach. I don't see at all how this question could take close to that long, so it's certainly not a question of 'too much math'. And if you want to do well on difficult absolute value questions, understand what absolute value measures: distance. Every high-level GMAT absolute value question I've seen is testing exactly that, and if you understand that well, these problems are not difficult.

[Stuart@KaplanGMAT]

My "too much math" alarm is sounding! I've always been a fan of the quick and dirty solution to GMAT questions.

If 3 < |x-5| < 7 , where x is an integer, how many possible values does x have?

That's all fine for this question, but what if you're asked:

What is the area of the region in the (x,y)-plane consisting of all points for which 3 <= |x-5| <= 7 and |y| <= 2 ?

Or:

What is the longest distance between two points which both satisfy:
3 <= |x-5| <= 7 and |y| <= 2

I think there's a lot of value in learning how to work out the values of x which satisfy the absolute value equation, at least if you're aiming for a top Quant score.

What I'm really saying is that if you're aiming for a 90th+ percentile in GMAT math, turn your 'too much math' alarm off- you're going to need to do some math. It's when the math would take longer than two minutes that you should look for another approach. I don't see at all how this question could take close to that long, so it's certainly not a question of 'too much math'. And if you want to do well on difficult absolute value questions, understand what absolute value measures: distance. Every high-level GMAT absolute value question I've seen is testing exactly that, and if you understand that well, these problems are not difficult.

Well, I guess we have a very different approach to the GMAT.

My philosophy is that the GMAT is really about strategy and critical thinking. It's certainly important to know the math, but an in depth knowledge of how the GMAT works is what's going to get you a fantastic score on test day. Anyone can learn textbook approaches (and there are certainly times when the textbook approach is the best way to answer a question) - a lot of people on this board have very strong math skills. The key to GMAT success is to really understand the concepts underlying the math so that you can find creative solutions to problems.

For example, I'm very good at math. However, I find that alternative approaches are very frequently quicker for me. Do I understand how to solve for absolute values? Sure. Is solving for absolute values the quickest way to solve every absolute values question? Definitely not.

So, when I post a suggested solution to a problem on these boards, if someone else has already detailed how to do the algebra, I'm going to offer an alternative approach. You posted the textbook solution, I showed that on this question, there are other ways one could proceed that would in fact be much quicker. A common GMAT "trick" is to make questions much more complicated than they actually are and test takers who can boil those questions down to their essence are going to bank time for the questions that require brute forcing some math.

In my experience, strong mathematicians who use textbook approaches on every GMAT question run out of time. Strong mathematicians who aren't wedded to the textbook approach and who know when to do the math and when not to are the ones who get fantastic scores on test day.

It's always been the Kaplan philosophy that there's no "correct" way to answer a question; the more tools you have at your disposal, the more likely it is that you'll pull the best ones out on test day.

[Ian Stewart]

Well, I guess we have a very different approach to the GMAT.

No, not at all- from what you say above, we have a very similar approach. In particular, this:

The key to GMAT success is to really understand the concepts underlying the math so that you can find creative solutions to problems.

really does capture what is essential on test day.

I'd take issue with only one point:

So, when I post a suggested solution to a problem on these boards, if someone else has already detailed how to do the algebra, I'm going to offer an alternative approach. You posted the textbook solution,

The textbook solution is the algebraic one. I posted a two-line solution based on the distance interpretation of absolute value- not the textbook solution, no algebra, and it takes ten seconds to do, if you understand the underlying concept. I like your solution as well- also very fast, and there's a lot of value in seeing different solutions to these problems. I posted what I did only because I saw the 'too much math' comment after my post, and I would not want people reading this thread to think there was 'too much math' involved when interpreting absolute value as a distance; indeed, if you're aiming for a 50 or 51 on the quant section, and you see an absolute value question on your test, you're almost certainly going to need to understand that |x-5| measures the distance between x and 5.