Is xm < ym ? (1) x > y (2) m < 0

Target question: Is xm < ym?
Another approach is to rephrase the target question.

Take the inequality and subtract ym from both sides to get: xm - ym < 0
Factor out the m to get m(x - y) < 0
In other words....
REPHRASED target question: Is m(x - y) a NEGATIVE value?

Statement 1: x > y
Subtract y from both sides to get: x - y > 0
So, x - y is a POSITIVE value
Does this provide enough information to determine whether or not m(x - y) a NEGATIVE value?
No. We still don't know anything about the value of m
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: m < 0
In other words, m is NEGATIVE
Does this provide enough information to determine whether or not m(x - y) a NEGATIVE value?
No. We don't know anything about the value of (x - y)
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x - y is a POSITIVE value
Statement 2 tells us that m is a NEGATIVE value
So, m(x - y) = (NEGATIVE)(POSITIVE) = some NEGATIVE VALUE
In other words, m(x - y) is definitely a NEGATIVE value
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer:

RELATED VIDEO FROM OUR COURSE

Using both statements, if we take the inequality in Statement 1:

x > y

and multiply by m on both sides, we need to reverse the inequality, because, as Statement 2 tells us, m < 0. So we get

xm < ym

which is what we wanted. Statement 1 is not sufficient alone because we need to know if m is positive or negative, and Statement 2 is not sufficient alone since we have no information about x or y. So the answer is C.

this is more simple method

xm<ym
move ym to the left
xm-ym<0
m(x-y)<0

now we can analyse , applying case 1 and 2.

In this case, we can subtract ym from both sides even though we don't know their signs?

We are concerned about the sign of a variable when multiplying/dividing an inequality by it. However we can safely add/subtract a variable from both sides of an inequality regardless of its sign.

9. Inequalities

• Theory
  Inequalities Made Easy!
  Inequalities: Tips and hints
  Arithmetic with Inequalities
  Wavy Line Method Application - Complex Algebraic Inequalities
  Solving Quadratic Inequalities: Graphic Approach
  Inequalities - Quadratic Inequalities
  Graphic approach to problems with inequalities
  Inequalities trick
  INEQUATIONS(Inequalities) Part 1
  INEQUATIONS(Inequalities) Part 2
  
  • Questions
    Inequality and absolute value questions from my collection
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

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When it comes to solving inequalities, we can safely add or subtract ANY value from both sides, and the resulting inequality will be perfectly valid.
However, if we multiply or divide both sides by a NEGATIVE value, then we must REVERSE the direction of the inequality symbol.

More here:

Hi!! Is this approach correct?

xm<ym
xm-ym<0
m(x-y)<0
is m<0 and x<y?

1. x>y but does not tell us the value of m hence not sufficient

2. M>0 but does not tell us for x or y hence not sufficent.

Combining 1+2, we get x>y and M>0 hence a definite yes.

Bunuel
VeritasPrepKarishma

Evaluate this properly: m(x-y)<0
This means the product of m and (x - y) is negative. So, either m is positive and (x - y) negative OR m is negative and (x - y) positive.
The two statements together tell us hat (x - y) is positive and m is positive.

So we know for sure that m(x-y) is positive. Sufficient.

Hello! Could someone please help me understand where my logic is flawed?
My initial answer was E
Where does the assumption that x and y are positive come from?
If x is -2 and y is -3, they still satisfy (1) x>y equation. Is x,y are positive e.g. x = 3 and y = 2, they also satisfy (1) x>y.
From (2) we know that m<0, so m could be -1.
So how can we tell for sure if xm<ym if x,y can be both positive and negative?

We certainly cannot assume x and y are positive here, so if any post above did that, that solution was not correct.

But if you look at your two examples:

• if x = 3, y = 2, and m = -1, then xm = -3, and ym = -2, so xm < ym, and the answer to the question is "yes"
• if x = -2, y = -3, and m = -1, then xm = 2 and ym = 3, so xm < ym, and the answer to the question is again "yes"

and you'll continue to find the answer to the question is "yes" for any valid examples you try, using both Statements. You can also prove the answer must be C here conceptually, as I did earlier in the thread.

Solution:

We need to determine whether xm < ym.

Statement One Alone:

Statement one alone is not sufficient since we don’t know the value of m. For example, if x = 2, y = 1 and m = -1, then we do have xm < ym. However, if x = 2, y = 1 and m = 1, then we don’t have xm < ym.

Statement Two Alone:

Statement two alone is not sufficient since we don’t know the values of x and y. For example, if x = 2, y = 1 and m = -1, then we do have xm < ym. However, if x = -2, y = -1 and m = -1, then we don’t have xm < ym.

Statements One and Two Together:

Both statements together are sufficient. If we multiply the inequality x > y by m, which is negative, we need to switch the inequality sign and obtain xm < ym.

Answer: C
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That's how you should approach such questions of Quantitative Reasoning.

Wondering if the answer should be B instead?

From statement 2 we already know m is negative, so we can divide both sides of the inequality by m (reverse the sign), and get x > y?

It's important in DS to distinguish between facts and questions. When we use only Statement 2 here, we know only one fact: m < 0. The inequality "Is xm < ym?" is not a fact -- that's a question, so we don't know if it's true. We can use Statement 2 to rewrite that inequality -- as you point out, we can divide by m on both sides, and since m is negative, we will flip the inequality. But when we do that, we aren't proving anything. All we are doing is rephrasing the question, hopefully in a way that makes it easier to answer. So when we use Statement 2 alone, the question becomes "Is x > y?", but we have no way to know if that's true without more information.

Notice though how useful rephrasing the question can be -- we just learned that the thing we want to know is if x > y is true, and that's precisely what Statement 1 tells us, so we immediately see that we can answer the question using both Statements.

Great explanation BrentGMATPrepNow. One question can we rephrase m(x - y) < 0 to m<0 ? x<y ? Thanks Brent

No we can't do that.
In general, if xy < 0, then EITHER x < 0 and y > 0, OR x > 0 and y < 0

So, you COULD (although I don't think it would help you) take the (already-rephrased) target question, "Is m(x - y) < 0" and rephrase it as "Is EITHER m < 0 and (y - x) > 0, OR m > 0 and (y - x) < 0?"

Great explanation and noted. Thanks BrentGMATPrepNow

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