This is QID 1598 in the GMATFix Solutions Engine. Follow the link for a more thorough discussion.

In brief:

ax and b must have opposite signs because they add up to 0. We'd like to know whether x > 0.

Statement 1 a+b > 0
This gives no restriction whatsoever on the sign of either variable, so it cannot be helpful.

Statement 1 is not sufficient

Statement 2 a-b > 0
Same as statement 1. We can come up with any sign for either variable, so this cannot be helpful because it doesn't provide any info about the signs.

Statement 2 is not sufficient

Together
We can add the two inequalities since the symbols face the same direction. adding a+b>0 to a-b>0 gives us 2a>0 --> a>0.

So we know that a is positive. Look back at the original equation (ax+b=0) and remember that it means that ax and b have opposite signs.

Since a is positive, ax could be >0 and b <0, which would make x > 0
Since a is positive, ax could be <0 and b > 0, which would make x < 0

We do not have a definitive answer (x could be positive or negative). In fact, x could equal 0 if b also equals 0.

Together the statements are not sufficient.

Answer: E

[/u]

Let's take the original equation and rewrite it as x = -(b/a). We'll know the sign of x if we can figure out the sign of (b/a).

(1) tells us that a > -b. So what!
(2) tells us that a > b. Again, so what!

(1) + (2) allows us to add the two equations together, like so:

(a > -b)
+ (a > b)
--------
2a > 0

So a is positive, and x = -(b/a) = -(b/positive). So the sign of b will give us the sign of x ... but we don't know the sign of b, so we can't answer the question.

If ax + b = 0, is x > 0 ?
a) a + b > 0
b) a - b > 0

Given that ax + b = 0, if a=0, then b=0.
Looking at the statements, we can see that it is not possible that both a=0 and b=0.
Thus, we know that a≠0, allowing us to rephrase the question stem.

ax + b = 0
ax = -b
x = - (b/a).

Substituting -(b/a) = x into x > 0, we get:
-(b/a) > 0
b/a < 0.

Question stem rephrased: Do a and b have different signs?

Both statements are satisfied by a=10 and b=1.
In this case, a and b have the same sign.
Both statements are satisfied by a=10 and b=-1.
In this case, a and b have different signs.
Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.[/quote]