APPROACH #1: Inequality property
Given:
x > 1
y > -1

If two inequalities are such that their inequality symbols are facing the same direction, we can add the two inequalities.
In this case we get: x + y > 1 + (-1)
Simplify to get: x + y > 0
From here we can subtract x from both sides to get y > -x, which can be rearranged to also get -x < y
Answer: D

APPROACH #2: Process of elimination
If x > 1 and y > -1, then it could be the case that x = 2 and y = 1
When we plug x = 2 and y = 1 into each answer choice, we get:
(A) (2)(1) > -1 TRUE
(B) (2)(1) < -1 FALSE - ELIMINATE
(C) -2 > 1 FALSE - ELIMINATE
(D) -2 < 1 TRUE
(E) 2 < 1 FALSE - ELIMINATE

So, we can already see that the correct answer is either A or D.
Let's test another pair of values.
If x > 1 and y > -1, then it could also be the case that x = 4 and y = -0.5
When we plug x = 4 and y = -0.5 into each remaining answer choice, we get:

(A) (4)(-0.5) > -1 FALSE - ELIMINATE
(D) -4 < -0.5 TRUE
By the process of elimination, the correct answer must be D