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:

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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.

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.
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Hi,

m(x-y)<0 -- > m<0 or x>y
do we need 2 conditions together or each condition alone is suff??

as it is m<0 OR x>y can we not select D as the answer?
please clarify.

Thanks.

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:

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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.
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Answer:

1) Given: x > y,
now if m is positive, multiplying both sides with m will not change the sign, thus xm > ym
but if m is negative, multiplying both sides with m will change the sign, thus xm < ym

so both cases are possible depending on the value of m. NOT SUFFICIENT.

2) Given: m <0, clearly NOT SUFFICIENT as not information about x and y is given.

Combining 1) and 2)
from two we know that m<0, hence out of the two cases in option 1, only this 'xm < ym' remain true. so BOTH STATEMENTS COMBINED ARE SUFFICIENT.

Can't I take out the M from both sides by multiplying with 1/m ? That leaves x<y.. hence I chose A. Can someone clarify

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An important rule of Inequalities:

If m> n. Is m+k>n+k.
Yes Always

If m>n, Is mr>nr?
Yes if r>0 No if r<0

Don’t do that when it comes to Inequality questions .

Because , the constant m is going to decide the sign of Inequality ,

If m is -ve , then multiplying both sides (left hand side and Right hand side ) will change the sign of Inequality .

Hope this helps

Is \(xm < ym\) ?
Is \(xm - ym < 0?\)
Is \(m (x-y) < 0?\)
Is \(m > 0\) and \(x < y\) or \( m < 0\) and \(x > y\)?


(1) \(x > y\)
No information about m. INSUFFICIENT.

(2) \(m < 0\)
No information about x or y. INSUFFICIENT.

(1&2) Combined we have the second scenario: \( m < 0\) and \(x > y\)

SUFFICIENT.

Answer is C.
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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.
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