If r + s > 2t, is r > t ?

(1) t > s

(2) r > s


Looking at the solution, it appears the inequalities are just added to each other and solved. Do inequalities work the same way as regular equations? For instance, if you have:

X - Y = 5
X + Y = 7

You get: 2X = 12


Can you do the same thing with inequalities?

For instance, in the equation above:

r + s > 2t
t > s

You get: r + s + t > 2t +s
Therefore: r>t

Can someone please confirm this for me?

My initial reaction was not to add inequalities as the solution did. Rather, I wanted to plug in numbers and identify under what circumstances each equation would work - if at all (for instance, I checked values of negative s, and negative t....compared the values.....for the second equation "r>s" - since there was no relation to t provided, I assumed that it was insufficient....which was obviously incorrect)

Any thoughts?

Thank You

If r + s > 2t, is r > t ?

(1) t > s

(2) r > s


1) t>s <=> t-s>0
r+s > 2t =>
r > t+(t-s) where t-s>0 =>
r>t SUFFICIENT

2) r>s =>
r+r>r+s
2r>r+s>2t
r>t
Sufficient

Answer D.

We should not add/subtract/multiply/divide variables to the inequalities without knowing whether the integers presented are non-zero or non-negative. I feel the answer is still (D).

Given r + s > 2t

Statement (1) says t > s which means s < t

So the given inequality r + s > 2t can be re-written as

r + (<t) > 2t ==> This implies r has to be greater than t to satisfy the inequality r + s > 2t.

SUFFICIENT.

Statement (2) says that r > s which can be re-written as s < r

So the given inequality can be re-written as r + (<r) > 2t ==> this also implies that r has to be greater than t.

SUFFICIENT.

So the answer is (D).

You can, indeed, add inequalities AS LONG AS the signs are identical and pointing in the same direction. And, in fact, when we can do this, it is often the easiest way to handle the math.

Trying numbers is a great fall-back method, and on some problems it should be the main method, but the one drawback is that we might not try all the right numbers or find the circumstances we need to prove or disprove something. So just have to be careful there.

Separately, for inequalities, you do not want to multiply or divide by variables if you do not know whether those variables are positive or negative.

Stacey,

In the above post you have mentioned that we can add or substract ineqalities, I have read other of your posts and you have mentioned that we can add but not substract them. So I was just making sure that this adding is allowed and substracting not allowed and that this was just by mistake

Oops - yes, it should've just said "add" not "add or subtract." Sorry about that! I've edited the post.