DS - Inequality

[achieve_dream]

If 2p != -q, is ((2p-q)/(2p+q)) > 1?

(1). p < 0
(2). q > 0

I did this:
2p-q > 2p+q

Cancelled like terms on left and right sides of inequality.

=> 0 > 2q => q < 0

and concluded that by knowing the sign of q, I will be able to give answer the question.

But I was wrong and should not have cancelled the like terms on on left and right sides of inequality.

Can experts help me to decide when to cancel terms and when to not cancel terms on left and right sides of inequality?

[Geva@EconomistGMAT]

If 2p != -q, is ((2p-q)/(2p+q)) > 1?

(1). p < 0
(2). q > 0

I did this:
2p-q > 2p+q

Cancelled like terms on left and right sides of inequality.

=> 0 > 2q => q < 0

and concluded that by knowing the sign of q, I will be able to give answer the question.

But I was wrong and should not have cancelled the like terms on on left and right sides of inequality.

Can experts help me to decide when to cancel terms and when to not cancel terms on left and right sides of inequality?

Combining like terms is fine - the problem isn't there, but rather in the initial transition from ((2p-q)/(2p+q)) > 1?
to 2p-q > 2p+q in the first place. What you are doing is multiplying the inequality by 2p+q. The problem is that you can't multiply an inequality by a unknown without knowing the unknown's sign: If sp+q is positive, then you can multiply without any changes, BUT if 2p+q is negative, you would need to flip the inequality sign because of multiplying by a negative number. Thus, without any indication of whether wp+q is positive or negative, you can't multiply because you don't know what to do with the sign.

This is in fact the issue of the question. Given that p<0 and q>0, the numerator of the fraction (2p-q) will always be negative. However, the denominator 2p+q can be positive or negative, depending on the value of p and q.
if 2p-q is positive, then the fraction is pos / neg will definitely be <1, and the answer is "no"
If 2p-q is negative, then (2p-q)/(2p+q) will be neg / neg = pos. Not only that, but the numerator will be "more negative" than the denominator (greater in absolute value), so the fraction will end up >1, with an answer of "yes". Thus, the answer is E.

Main takeaway from this question: Do not multiply or divide an inequality by an unknown, unless you know the unknown's sign.

[achieve_dream]

Thankyou Geva.
BTW, the official answer is also E as Geva said.

Source: Veritas Prep Practice test