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