Is t =|r - s|? (1) 3r > 3s (2) t = r -s

(1) Nothing about t NS

(2) t = r-s. It is tempting to mark sufficient, but lets test some values Let r=-2 t=-3 then t =-2--3 =1 and |r-s| - |1| =1 Yes

let r=-2, s=1 then t =-2 - 1 =-3 and |r-s| =|-3| = 3 NS


(1) and (2) 3r>3s --> r>s And t=r-s Logic tells us that t will always be positive thus suff, however we can test some values just to be sure.

Notice for values of r and s r >S So, let r=-3, s=-4 then r-s = -3 --4 = 1 and |r-s| = |1| =1 Yes

Let r=-1/2 and s=-1 then r-s =1/2 and |r-s| =1/2 Yes

Let r=-1/2 s = -3/4 then r-s = 1/4 and |r-s| = |1/4| = 1/4 Yes

Let r = 1/2 s =1/4 then r-s = 1/4 and |r-s| = 1/4 Yes

Let r=2 and s=1 then r-s = 2-1 = 1 and |r-s| =|1| = 1

So whether we pick negative integers, negative fractions, positive fractions, or positive integers for r and s we get a Yes answer we can mark sufficient

C

Manat,

the question asks if t =|r-s|.. note that the modulus is always non-negative. So, we can answer this question as yes or no, based on knowing if t = r-s or t = s-r depending on which of r & s is bigger.

For example, if r > s and t = r - s then we can say that yes, t = |r - s|, but it would be a certain "no" if r < s. Here, r - s would be negative and t cannot take a negative value.

Let us look at -

Statement (1)
3 r > 3s
or,
r > s ...does not give us any info to relate with t, hence insufficient

Statement (2)
t = r - s
This too is insufficient as depending on which one is bigger, r - s could either be negative or positive and hence we cannot be sure if t = |r - s|

Combining the two together - we can definitely say that t = |r-s| and that r-s is positive.

Hope this helps

@Gladiator59:- Crystal clear. Thank you.

What's the source for this question?

Hi dcwanderer30,

Apologies as I don't know the source of above question as someone posted it in https://gmatclub.com/chat study group and it was a nice question hence shared with this forum.

IanStewart Can you help with if this question were to be written differently in two alternative ways as follows?

Alternative question 1:
Is t = |r-s| ?
1) 3r > 3s
2) t = - (r-s)

Alternative question 2:
Is t = |r-s| ?
1) 3r < 3s
2) t = - (r-s)

If you just think about two numbers, like 3 and 7, then when we subtract these two numbers in either order, we get either -4, the negative difference of 3 and 7, or we get 4, the positive difference of 3 and 7. If we take the absolute value after we subtract, though, we always get the positive difference: |3 - 7| = |7 - 3| = 4. That's also the distance between 3 and 7 on the number line. Similarly, when we have two different unknowns r and s, then r-s might be the positive difference of r and s, or it might be the negative difference of r and s. But when we take the absolute value, |r - s| is always the positive difference of r and s. So when a question asks "Is t = |r - s|?", the question is asking "Is t equal to the positive difference of r and s?".

I'll simplify your two Alternative Questions in the obvious ways:

Qn 1:
Is t = |r - s|?
1. r - s > 0
2. t = s- r

Here, Statement 2 tells us "t is either the positive or negative difference of r and s". We don't know which. So we can't answer the question. Using both Statements, we know r-s is the positive difference of r and s. So Statement 2 tells us t is equal to the negative difference or r and s, and if t is negative, there's no chance it equals an absolute value of any kind, let alone the positive difference of r and s, so the answer here is C; using both Statements, we can be certain the answer to the question is 'no'.

Qn 2:
Is t = |r - s|?
1. s - r > 0
2. t = s - r

Similarly, Statement 2 tells us "t is either the positive or negative difference of r and s". That's not sufficient alone, but Statement 1 ensures s-r (not r-s) is the positive difference of s and r, so using both Statements, the answer to the original question must be "yes", and the answer is C. Algebraically, if we know Statement 1 is true, we know s - r = |s - r| = |r - s|, so if t = s - r, then t = |r - s|.

If you wanted to test numbers instead, you'd just need to experiment with two scenarios: one where s > r, and one where r > s, and you'd also find C is the answer to each question.

IanStewart

I have a related question regarding this type of inequality problem. What if we're asked if (y+5)(y-3) > 0? and given the following statements

1) y > 3
2) y < -5

Since y > -5 and y > 3 or y < -5 and y < 3, would neither statement be sufficient?

The product (y+5)(y-3) will be positive in one of two cases:

- when both factors are positive. Notice that y-3 is always less than y+5 (it's exactly 8 less, no matter what y is), so as long as y-3 is positive, y+5 always will be. So the product will be positive if y - 3 > 0, or if y > 3

- when both factors are negative. Similarly, if y+5 is negative, y-3 will certainly be negative, because it's smaller than y+5. So both factors will be negative if y < -5.

So each statement is sufficient alone in your question, and the answer is D.

edit: and I should have added that the statements are contradictory, as Bunuel points out below, so this is not a possible GMAT question, though you could see either statement alone in a question like this (along with another consistent statement).

I'd just like to add one thing: this is a flawed question by GMAT's standards. On the GMAT two statements do not contradict each other, while y > 3 and y < -5 clearly do.

Got it, thanks for pointing out the flaw! What if the question was (y+5)(y-3) < 0 instead and given the following statements:

1) y > -5
2) y > 3

I'm thinking there are two scenarios: A. -5 < y < 3 and B. y < -5 and y > 3. However scenario B is not possible.

Would we be able to determine sufficiency? IanStewart Bunuel any chance either of you could weigh in on this?

IanStewart Was hoping to follow up to see if you could help with a modified question since the previous one was flawed. Thank you so much!

What if the question was (y+5)(y-3) < 0 instead and given the following statements:

1) y > -5
2) y > 3

I'm thinking there are two scenarios: A. -5 < y < 3 and B. y < -5 and y > 3. However scenario B is not possible.

Would we be able to determine sufficiency?

Your scenario analysis is correct; the inequality (y+5)(y-3) < 0 is true only when -5 < y < 3. So if a DS question asks "is (y+5)(y-3) < 0?", we can rephrase that question more simply as "Is -5 < y < 3?" Statement 1 then is not sufficient; while Statement 1 guarantees that y > -5, we have no way to know whether y < 3 (y might be zero, or might be one million, so the answer can be yes or no). Statement 2, however, is sufficient, because if y > 3 is true, we know with certainty that y is not between -5 and 3, so we can be certain the answer to the question is 'no'.

Got it, thank you so much IanStewart!

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