If x and y are integers, and x - y + 2 > 0, is x negative?

Given: x - y + 2 > 0

Target question: Is x negative?

Statement 1: x + y > 0
There’s a useful inequality property that says If two inequalities have their inequality symbols facing the same way, we can ADD the inequalities.

We have:
x - y + 2 > 0
x + y > 0

When we add the two inequalities we get: 2x + 2 > 0
Subtract 2 from both sides of the inequality: 2x > -2
Divide both sides by 2 to get: x > -1
Since x is an integer, we know that x could equal 0 or 1 or 2 or 3 or 4 or . . . .
For all possible values of x the answer to the target question is, NO, x is not negative
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 2x - 1 < y
We first need to rearrange this inequality so that the inequality symbol is facing the same direction as it is in the given information
We can first rewrite it as: y > 2x - 1
Now subtract 2x from both sides to get: y - 2x > -1
We can rearrange the terms to get the following: -2x + y > -1

We now have:
x - y + 2 > 0
-2x + y > -1

When we add the inequalities we get: -x + 2 > -1
Subtract 2 from both sides of the inequality: -x > -3
Multiply both sides of the inequality by -1 to get: x < 3 (since we multiplied both sides of the inequality by a NEGATIVE value, we must reverse the direction of the inequality symbol)
If x < 3, then x COULD equal 2, in which case the answer to the target question is, NO, x is not negative
Conversely, x COULD equal -1, in which case the answer to the target question is, YES, x is negative
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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