Data Sufficiency

Hey Guys..

I have a simple question here.

If x>1,what is the value of integer x?
(1)There are x unique factors of x.
(2)x is a prime number that lies between 6 and 8.

Its a straight forward question.Since its a value question,are'nt the two statements supposed to give the same answer?I mean both statements are sufficient to get to the answer,but they provide different answers.Is this possible on the GMAT?Because I thought that the statements always complemented each other.

Also when dealing with data sufficiency questions,is it better to 1.Look at the question stem and then look at just one statement 1 first up and check for sufficiency,If that statement doesnt work,then we look at question stem again,and have a good hard look at statement 2?Is this technique good?

Consider the following problem.I have deleted the statements.

If x>1,what is the value of integer x?

Now lets assume we have 2 statements 1 and 2,and a term x in each of those

When we test a particular value for statement 1 obeying x>1 ,is it necessary to test the same value for statement 2?Or can we test anyother value as long as the basic condition of x>1 is met.

Agree completely with Rich!

To your first point the two statements must have some value in common at all times and the only way to get answer choice D is to have the same answer to both statements. So if statement 2 points to 7 as the only possible value then statement 1 must ALLOW for 7 as well. If it does not allow for 7 then whoever wrote the question is mistaken (as is the case here). If it does allow for 7 as well as some other numbers then that statement is not sufficient and the answer is B (only statement 2 is sufficient with the specific value of 7).

I wrote an article about this very subject! Please read it... https://www.beatthegmat.com/mba/2012/01/ ... ufficiency

As to your second point - Rich is right on this one too. You absolutely want to "reuse" numbers that worked from a previous statement. There are two reasons for this: 1) you do not have to think of new numbers and 2) much more importantly, if you do end up taking the two statements together (as in C versus E) you might already have a value that works for both statements.

David,I have a doubt about the procedure used here.
Is x ≥ 0?

1) x^2 = 9x

2) |x| = -x

Going back to the article,Statement 1 is absolutely sufficient.When it comes to statment 2,this is my way of solving it and it contradicts with the answer you've provided.

|x|=-x =>x=+(-x)=-x,this is true when x>0, and x=-(-x),this is true when x<0
Therefore statement 2 gives us 2 conditions x=x and x=-x.Now since x=-x when x>0,we cant test 0 there.Now,we also have x=x when x<0,so -5=-5,-6=-6,these dont satisfy the condition in the question stem and thus its a NO.Thus the answer is D.

In which way is my understanding of the question wrong?

I have followed this technique from here--
https://www.manhattangmat.com/strategy-s ... -value.cfm

Absolute value expressions start to become difficult when variable expressions are placed inside the bars. For example, /x/. Upon a cursory examination, the expression /x/ seems like it should be equal to x. Since there is no sign in front of the x, the absolute value bars should be able to be removed without jeopardizing the "guarantee of positive." What this line of reasoning fails to account for, however, is that x itself could be negative! When dealing with absolute value expressions that contain variables, two scenarios must be considered: (1) the scenario whereby the expression inside the bars is positive and (2) the scenario whereby the expression inside the bars is negative.

In this example, for scenario (1) if x > 0, the expression /x/ can simply be represented as x; for scenario (2) if x < 0, the expression /x/ must be represented as (-x). Notice that in the negative scenario, we don't simply remove the absolute value bars. We remove the absolute value bars and negate the entire expression within.

Let's look at a more complicated example: the expression /x - 3/. As always, we must consider both the positive and negative scenarios. When is the expression inside the absolute value bars positive? Not simply when x > 0, but when x - 3 > 0 or when x > 3. Likewise the expression will be negative when x < 3.

To recap, the two scenarios are:
(1) /x - 3/ can be rewritten as x - 3 when x > 3
(2) /x - 3/ can be rewritten as -(x - 3) or 3 - x when x < 3

One more for the road: /3x + y/.
(1) /3x + y/ can be rewritten as 3x + y when 3x + y > 0
(2) /3x + y/ can be rewritten as -(3x + y) when 3x + y < 0


Thanks David.Brilliant!!

David@VeritasPrep wrote:Agree completely with Rich!

To your first point the two statements must have some value in common at all times and the only way to get answer choice D is to have the same answer to both statements. So if statement 2 points to 7 as the only possible value then statement 1 must ALLOW for 7 as well. If it does not allow for 7 then whoever wrote the question is mistaken (as is the case here). If it does allow for 7 as well as some other numbers then that statement is not sufficient and the answer is B (only statement 2 is sufficient with the specific value of 7).

I wrote an article about this very subject! Please read it... https://www.beatthegmat.com/mba/2012/01/ ... ufficiency

As to your second point - Rich is right on this one too. You absolutely want to "reuse" numbers that worked from a previous statement. There are two reasons for this: 1) you do not have to think of new numbers and 2) much more importantly, if you do end up taking the two statements together (as in C versus E) you might already have a value that works for both statements.

Thanks Rich.Awesome!!

[email protected] wrote:Hi dddanny2006,

In answer to your first question: YES, if both statements are Sufficient, then they will both have the SAME answer.

This question either has a typo in it or the person who wrote it didn't understand how the GMAT writes its questions.

In answer to your second question: After reading the prompt, you're supposed to deal with each Fact INDIVIDUALLY. Once you've done whatever work is necessary to deal with Fact 1 by itself (TEST Values, Do math, Use Number Properties, Use basic logic, etc.) and have determined that it is either Sufficient or Insufficient, THEN you deal with Fact 2 (again, by itself).

If I'm TESTing Values, I certainly look to see if any of the Values that I used in Fact 1 will "fit" with Fact 2. Since we're supposed to deal with each Fact individually though, I consider what Fact 2 gives me to work with and I look for the various possibilities given that info.

GMAT assassins aren't born, they're made,
Rich

You have applied the portion in bold italics incorrectly. The portion talks about x> 0 and x < 0 but it does not talk about x = 0. That is because x= 0 is far simpler.

You can certainly test zero. I looked this up in a dozen reputable locations just to be 100% sure before I wrote the article. Every source agrees that |0| = 0.

As one source says "absolute value is distance from zero. How far is zero from zero? That distance is zero so that is what the absolute value of zero equals."

So basically we have 3 conditions for |x|=-x,?
The first one is x=-x when x>0 ---Is this correct?The x>0 is correct here,isnt it?Or am I wrong here?
The second is x=-(-x)=x when x<0------Is this correct too?
Is that x>0 and x<0 a wrong concept?Or is that a concept thats not required and all whats required are those 2 equations?

Whats that third one like?How do we rephrase it when x=0?I was of the opinion that |x| gives us 2 conditions.

Also,lets look at this condition--

x=-x when x>0 We know that x can never be equal to -x when its greater than 0. For example 1!=-1
Another condition, x=x when x<0 -1=-1 .Therefore its a NO to the question stem because -1 is not greater than or equal to Zero.

In which of these equations do we test a 0? x=-x or x=x? Either of them work well,but the conditions x>0 and x<0 dont do any good here.
Assuming we still proceed and substitute 0.We get a YES to the question stem.Since we have both a Yes and a No, Statement 2 is insufficient.

Going back to the italicized statement.x=-x when x>0 ,how do we go about this condition?We got to get x a value that equals its negative too right,and then if we satisfy the equation we test x against the question stem right?What if we fail to get that value for the condition?Do we deem that particular condition of statement 2 as insufficient?In general not relating to this question assume we have a statement 2 that gives us 3 conditions.One of the 3 condtions doesnt work(assume x=-x when x>0),the other 2 conditions work well and when tested against the question stem result in 2 yes's and as a result statement 2 has 2 yes's and _________?What do I call that blank?A no or insufficient statement or what.What bearing does 2yes's and _______ have on the statement?Is that statement 2 insufficient or sufficient?
Why has that Manhattan-link article included the x>0 and x<0 conditions??

Im a little confused here David. Please help me understand.

David@VeritasPrep wrote:You have applied the portion in bold italics incorrectly. The portion talks about x> 0 and x < 0 but it does not talk about x = 0. That is because x= 0 is far simpler.

You can certainly test zero. I looked this up in a dozen reputable locations just to be 100% sure before I wrote the article. Every source agrees that |0| = 0.

As one source says "absolute value is distance from zero. How far is zero from zero? That distance is zero so that is what the absolute value of zero equals."

Great question! But don't let this get too confusing...by trying to memorize you are much more likely to get confused.

At Veritas we do not emphasize memorizing all of these small rules. The reason is that it is much easier to simply try a couple of numbers and prove it on the spot, during the test. You can do this in about 20 seconds if you need to.

Here we go.

if "|x| = -x" I know that positive and negative is the number property to focus on here.

Try a negative number. How about -5. If you put -5 into the absolute value it becomes |-5| which equals 5. If you put -5 into -x it becomes - (-5) which is also 5. Therefore negative numbers work. Absolute value makes a negative turn positive. Multiplying by a negative makes a negative turn positive. If you want to memorize something memorize those two things, since they are vital number properties rather than easily proven facts.

Now try a positive number. How about 9 since 9 worked for statement 1 in my example from the article. If you put 9 into absolute value it is |9| which equals 9. But if you put it into negative it becomes - (9) which equals -9. So positive numbers do NOT work here.

At this point you might not think of zero. That is the point of using this problem in my article. Statement 1) gives you the values of 0 and 9 for x. At this point statement 2) gives you all of the negative values as possible values for x. Yet there is no way for these statements to be true at the same time right? So that is the problem. You must have at least one value in common.

That is where zero comes in. We already know that statement 2 does not allow for positive numbers, like 9. But anything times zero is zero. So |0| = -0, since both equal zero.

This gives you the value that is acceptable to both statements. x = 0. However, you do not need to know specifically that x = 0, You only need to know if "x is greater than or equal to 0". Statement 1 gives you a consistent "yes." While statement 2 allows both a "yes, x = 0" and a "No, x is negative. And that is why the answer is A.

David,that's an awesome explanation.I needed to know where I went wrong in my method.Could you please tell me that.I have a Manhattan set and would like to stick to those methods and remembering too much stuff may get me too confused.Are those conditions x>0 and x<0 not necessary?Also you didnt answer the general question that I asked you.Here--In general not relating to the question assume we have a statement 2 that gives us 3 conditions.One of the 3 condtions doesnt work(assume x=-x when x>0),the other 2 conditions work well and when tested against the question stem result in 2 yes's and as a result statement 2 has 2 yes's and _________?What do I call that blank?A no or insufficient statement or what.What bearing does 2yes's and _______ have on the statement?Is that statement 2 insufficient or sufficient?

David@VeritasPrep wrote:Great question! But don't let this get too confusing...by trying to memorize you are much more likely to get confused.

At Veritas we do not emphasize memorizing all of these small rules. The reason is that it is much easier to simply try a couple of numbers and prove it on the spot, during the test. You can do this in about 20 seconds if you need to.

Here we go.

if "|x| = -x" I know that positive and negative is the number property to focus on here.

Try a negative number. How about -5. If you put -5 into the absolute value it becomes |-5| which equals 5. If you put -5 into -x it becomes - (-5) which is also 5. Therefore negative numbers work. Absolute value makes a negative turn positive. Multiplying by a negative makes a negative turn positive. If you want to memorize something memorize those two things, since they are vital number properties rather than easily proven facts.

Now try a positive number. How about 9 since 9 worked for statement 1 in my example from the article. If you put 9 into absolute value it is |9| which equals 9. But if you put it into negative it becomes - (9) which equals -9. So positive numbers do NOT work here.

At this point you might not think of zero. That is the point of using this problem in my article. Statement 1) gives you the values of 0 and 9 for x. At this point statement 2) gives you all of the negative values as possible values for x. Yet there is no way for these statements to be true at the same time right? So that is the problem. You must have at least one value in common.

That is where zero comes in. We already know that statement 2 does not allow for positive numbers, like 9. But anything times zero is zero. So |0| = -0, since both equal zero.

This gives you the value that is acceptable to both statements. x = 0. However, you do not need to know specifically that x = 0, You only need to know if "x is greater than or equal to 0". Statement 1 gives you a consistent "yes." While statement 2 allows both a "yes, x = 0" and a "No, x is negative. And that is why the answer is A.