Author 
Message 
TAGS:

Hide Tags

Senior Manager
Joined: 08 Dec 2015
Posts: 315

When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 10:16
Hi all,
nooby question: when can we manipulate an inequality?
as far as I understand we can : multiply\divide if we know that all variables are not zero and we always can add\subtract, so no need to know if the variables are diff from 0 or not.
However this rule of mine seems kinda incomplete..
Anything to add guys? I bet there is..
Thank you!



Current Student
Joined: 20 Mar 2014
Posts: 2683
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 10:31
iliavko wrote: Hi all,
nooby question: when can we manipulate an inequality?
as far as I understand we can : multiply\divide if we know that all variables are not zero and we always can add\subtract, so no need to know if the variables are diff from 0 or not.
However this rule of mine seems kinda incomplete..
Anything to add guys? I bet there is..
Thank you! Please use the search function for your question. It will provide you a better understanding of different topics on offer. As for inequalities, refer to this good collection of posts: inequalitiesmadeeasy206653.htmlManipulation of inequalities can be done only when you know for sure the sign of the variables in question as the inequality sign changes when you multiple (or divide) an inequality by a negative number. Try to add the inequalities with the SAME sign and not subtract. Example: x<3, y>5 , you can NOT add the 2 as the signs of inequalities are different. But you can multiple y> 5 by 1 to get, y<5 (flipped the inequality here as we were multiplying by a negative number) and now add to x<3 to get, xy<2. x<3 and y<5 > x+y <8 is perfectly valid. Hope this helps.



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 10:35
Thank you for your reply! Sorry, never found that link before! Gotta check it out :D



Manhattan Prep Instructor
Joined: 22 Mar 2011
Posts: 1190

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 12:13
Sometimes you can interpret a question that uses an inequality even if you don't know the signs. Consider the following question stem: Is x/y > 1? (There would probably be a disclaimer that y is not 0, but the GMAT isn't going to ask us about division by 0.) We can't multiply both sides by y without knowing if it's positive or negative, but we can look at the two cases. If y is positive, the question is now "Is x > y?" If y is negative, the question is now "Is x < y?" So now we have a twopart question. We need to know whether y is positive or negative. Now let's look at two statements: 1) x < y This answers the second part, but since we don't know whether y is + or , it doesn't give us a full answer. If y is positive, then this gives us a NO: x is not greater than y. If y is positive, we get YES: x is less than y. INSUFFICIENT 2) y is negative. By itself, this of course tells us nothing about x. INSUFFICIENT 1&2) Putting both statements, together, we see we're in the second case: y is negative, so we want to answer the question "Is x < y?" Statement 1 gives us a YES to that question. SUFFICIENT.
_________________
Dmitry Farber  Manhattan GMAT Instructor  New York
Manhattan GMAT Discount  Manhattan GMAT Course Reviews  View Instructor Profile  Manhattan GMAT Reviews



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 12:24
Wow that's a tricky one.. These things make me hate inequalities, when ones thinks it's all under control there is always something like this that shows up... Thank you so much for this example! Hope there aren't many like this on the actual GMAT



Manhattan Prep Instructor
Joined: 22 Mar 2011
Posts: 1190

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 12:34
I've seen a real GMAT q with a similar trick, but it's not the norm. However, you could probably nail this one by testing numbers without getting too theoretical. Just ask yourself "If x<y, does x/y have to be greater than 1?" Then you could test: If you plug in two positive integers that fit the statement, say x=2 and y=3, then you'll get a NO. x/y is not greater than 1. If you plug in two negative integers, such as x=3 and y=2, then you get a YES. Insufficient. 2) Again, you can test + and . I can get a YES by making both variables negative. If x is negative, I'll get a negative fraction, so that's a NO. 1&2) Now you're more limited. y is negative, and x has to be less than y. No matter what you plug in, x/y will always be > 1. For instance, if y=2 and x=3, then x/y=3/2. If y=4 and x=100, then x/y=25. SUFFICIENT
_________________
Dmitry Farber  Manhattan GMAT Instructor  New York
Manhattan GMAT Discount  Manhattan GMAT Course Reviews  View Instructor Profile  Manhattan GMAT Reviews



Current Student
Joined: 20 Mar 2014
Posts: 2683
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 12:36
iliavko wrote: Wow that's a tricky one.. These things make me hate inequalities, when ones thinks it's all under control there is always something like this that shows up... Thank you so much for this example! Hope there aren't many like this on the actual GMAT What did you understand when the question asked you "is x/y > 1"? In what cases will you get a yes and in what cases will you get a no? From the first glance, x/y >1 (if its true) should tell you: 1. x and y MUST be of same sign as only then a fraction is >0. Think : negative/negative or positive / positive) 2. If x/y > 1 > the absolute value of x > absolute value of y > x > y > this statement when taken in isolation should mean that when y<0, x<y and when y>0 , x>y. If you dont get this straightaway, just test a few cases out. Build into such understanding and you will get better at quant.



Senior Manager
Joined: 08 Dec 2015
Posts: 315

When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 12:45
Dmitry and Engr,
Thank you very much for your input on this.
By default I don't like to test values, I use to do it a lot when I started, but if the stem doesn't have restrictions then you have tricky combinations to test like 1<y<0 and x>1, then reversed, then same sign, then both integers, then both fractions etc etc and for me 2 minutes would be up. So I always try to get the logic and then if I'm stuck go to testing values as a last resort.
This case here is very interesting from the abstract point of view, if you don't test values.
Ones again, thank you both!



Math Expert
Joined: 02 Sep 2009
Posts: 43810

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
26 Apr 2016, 22:35



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
27 Apr 2016, 07:10
Bunuel, thank you very much for the link. Just checked it, it has all the answers I needed!
:D



Manager
Joined: 18 Feb 2016
Posts: 102

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 08:23
I think the replies by DmitryFarber are sufficient.. I personally rely on Manhattan resources when prepared for my GMAT exam.



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 10:48
Imagine that we have a>0 b>0 c>0
\(\frac{a}{c}\)=\(\frac{d}{b}\)
So we know that a,b,c are positive but we know nothing about d
Are we allowed to crossmultiply b by \(\frac{a}{c}\) to get \(\frac{ba}{c}\)=d
(Assume that the expression in this form becomes useful to evaluate something) So basically can we multiply\divide the terms that we are sure to be positive even if there is another term we know nothing about? Or a situation like this is unlikely on GMAT? I am asking this to avoid not manipulating terms when I cold have.
Thank you!



Math Expert
Joined: 02 Sep 2009
Posts: 43810

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 13:29
iliavko wrote: Imagine that we have a>0 b>0 c>0
\(\frac{a}{c}\)=\(\frac{d}{b}\)
So we know that a,b,c are positive but we know nothing about d
Are we allowed to crossmultiply b by \(\frac{a}{c}\) to get \(\frac{ba}{c}\)=d
(Assume that the expression in this form becomes useful to evaluate something) So basically can we multiply\divide the terms that we are sure to be positive even if there is another term we know nothing about? Or a situation like this is unlikely on GMAT? I am asking this to avoid not manipulating terms when I cold have.
Thank you! Multiplying/dividing by positive/negative is relevant only for inequalities, when we are concerned about the sign of the inequality.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manhattan Prep Instructor
Joined: 22 Mar 2011
Posts: 1190

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 16:26
iliavko, if you meant that to be an inequality, then yes, that's a valid move. (Of course you could do that with an equation, too.) We only have to worry about flipping the sign when we actually multiply/divide by a negative. Let's take a look at an example: x/y < 1 If y is positive, then we can just move it over and get x<y. For x/y to be less than 1, x would have to be either a smaller positive number (yielding a conventional fraction) or a negative number (yielding a negative, which of course will always be <1). For instance, x could be 2 and y could be 3, or x could be 2 and y could be any positive number. If y is negative, we have to flip the sign when we move it over. x >y. Why is this so? Think about the kind of values you'd need to make it true. If we copy from our last example and just make the values negative, we could say x= 2 and y =3. This of course will give us 2/3 and satisfy the inequality. But notice that while 2<3, 2 is actually GREATER THAN 3. So basically, the whole reason we flip the sign is that negative numbers are counted backwards. The "smallest" ones are actually the biggest, making 1 much bigger than 100. Since the whole system runs opposite from the way we count positive numbers, whatever holds true on the positive side is reversed on the negative side. For that reason, when we don't know if a variable is positive or negative, we can't multiply/divide it to the other side, because we don't know if we need to switch scenarios. No other action is going to cause the same problem (unless we're unsquaring or removing and absolute value sign), so the other moves we know are safe.
_________________
Dmitry Farber  Manhattan GMAT Instructor  New York
Manhattan GMAT Discount  Manhattan GMAT Course Reviews  View Instructor Profile  Manhattan GMAT Reviews



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 16:44
That was supposed to be an inequality, sorry! Thank you very much for your explanations! Btw, assume that my example is an equation, so the signs wouldn't be important and we know that denominators are never 0. But would we have to know that the numerators are not 0 to crossmultiply? Could it be a trap? And look, sorry for these questions, but I am constantly discovering these "holes" in my math basics.. And it's tough to find the exact answer in a book or something



Manhattan Prep Instructor
Joined: 22 Mar 2011
Posts: 1190

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
28 Apr 2016, 22:17
No, we wouldn't have to worry about that. One easy way to think about it is that if a variable isn't "moving," then you don't need to know anything about its sign for the manipulation to work. You only need to worry about the elements you are actively manipulating. By the way, if one of the numerators in that equation did equal zero, then they would both have to. Since the denominators can't be 0, it would have to be both numerators to make the sides equal. So we could get something like ba/c = d or ba = cd, but it wouldn't really be giving us any information about b and c. Zero times anything is equal to zero.
_________________
Dmitry Farber  Manhattan GMAT Instructor  New York
Manhattan GMAT Discount  Manhattan GMAT Course Reviews  View Instructor Profile  Manhattan GMAT Reviews



Senior Manager
Joined: 08 Dec 2015
Posts: 315

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
29 Apr 2016, 07:32
Thank you very much for the clarifications, Dmitry!



NonHuman User
Joined: 09 Sep 2013
Posts: 13829

Re: When can we manipulate the Qstem in inequalities? [#permalink]
Show Tags
31 Jan 2018, 01:11
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: When can we manipulate the Qstem in inequalities?
[#permalink]
31 Jan 2018, 01:11






