If ax + b = 0, is x > 0

[BTGModeratorVI]

If ax + b = 0, is x > 0

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

Answer: E
Source: GMAT prep

[deloitte247]

$$T\arg et\ question\ =>\ is\ x>0?$$
$$ax+b=0$$
$$\frac{ax}{a}=\frac{-b}{a}$$
$$x=\frac{-b}{a}$$
$$Question\ =>is\ \frac{-b}{a}>0\ for\ x>0$$
$$b\ must\ be\ negative\ and\ a\ must\ be\ positive$$

Statement 1 => a + b > 0
The signs of a and b could be either +a, +b or +a, -b or -a, +b. The information provided was not enough to determine the signs of a and b so target question cannot be answered and statement 1 is NOT SUFFICIENT

Statement 2 => a - b > 0
The signs of a and b could be either +a, +b or +a, -b. The information provided was not enough to determine the signs of a and b, so the target question can not be answered and statement 2 is NOT SUFFICIENT

Combining both statements together => The signs of a and b could be either +a, +b or +a, -b. This information does not give the definite signs of a and b so the target question cannot be answered
Both statements combined together ARE SUFFICIENT

Answer = E

[Brent@GMATPrepNow]

If ax + b = 0, is x > 0

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

Answer: E
Source: GMAT prep

Target question: Is x > 0

Given: ax + b = 0

Statement 1: a + b > 0
At this point, we have 1 equation, 1 inequality, and THREE variables.
Even if we had 2 EQUATIONS and 3 variables, we probably wouldn't be able to make any conclusions about whether x is positive or negative.
Given this, let's TEST some values
There are several values of a, b and x that satisfy statement 1 (and the given equation ax + b = 0). Here are two:
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a - b > 0
This is the same scenario as statement 1, so let's TEST some values
There are several values of a, b and x that satisfy statement 1 (and the given equation ax + b = 0). Here are two:
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent