Is x - y r - s ? (1) x r and y

Question

Is x - y > r - s ?

(1) x > r and y < s
(2) y = 2, s = 3, r = 5, and x = 6

PS21276

Bunuel · Accepted Answer

Is x - y > r - s ?

(1) x > r and y < s. You can subtract inequalities if their signs are in the opposite directions (> <): x-y>r-s. Sufficient.

(2) y=2, s=3, r=5, and x=6. We have the exact values of each unknown, thus we can answer whether x-y>r-s. Sufficient.

Answer: D.

ADDING/SUBTRACTING INEQUALITIES:

You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

RAISING INEQUALITIES TO EVEN/ODD POWER:

A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
\(2<4\) --> we can square both sides and write: \(2^2<4^2\);
\(0\leq{x}<{y}\) --> we can square both sides and write: \(x^2<y^2\);

But if either of side is negative then raising to even power doesn't always work.
For example: \(1>-2\) if we square we'll get \(1>4\) which is not right. So if given that \(x>y\) then we can not square both sides and write \(x^2>y^2\) if we are not certain that both \(x\) and \(y\) are non-negative.

B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
\(-2<-1\) --> we can raise both sides to third power and write: \(-2^3=-8<-1=-1^3\) or \(-5<1\) --> \(-5^3=-125<1=1^3\);
\(x<y\) --> we can raise both sides to third power and write: \(x^3<y^3\).

For multiplication check here: help-with-add-subtract-mult-divid-multiple-inequalities-155290.html#p1242652

THEORY ON INEQUALITIES:

x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html
inequations-inequalities-part-154664.html
inequations-inequalities-part-154738.html

QUESTIONS:

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.

fozzzy · Answer

I didn't add the inequalities I manipulated the original question

Given x-y > r-s

We can than write this as x-r > y-s

FS 1 says X-r > 0 ( positive) and y-s<0 ( negative)

eeshajain · Answer

I'm unclear on how to select the sign when substracting the inequalities. Can someone explain?

Bunuel · Answer

Isn't that explained  here ? Let me know if anything is unclear in that post.

bumpbot · Answer

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Is x - y > r - s ? (1) x > r and y < s (2) y = 2, s = 3, r = 5, and

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