Ambiguity in Inequalities I was just going through an inequality and i couldn't make out where i was going wrong.
$(2+k)(1+2T)-5KT>0$
First approach 
$2+4T+K-3KT>0 \rightarrow T>-(2+K)/(4-3K) \rightarrow T>(2+K)/(3K-4)$
Second approach
$2+4T+K-3KT>0 \rightarrow 2+K>T(3K-4) \rightarrow T<(2+K)/(3K-4)$
which approach is correct and why?
Thank You
 A: If you do not know anything about $K$ a priori, none of them are correct (or none of them are incorrect).
Let me explain myself. When you are dealing with an inequality, you cannot divide both sides of it by a quantity unless you are sure of the sign of it. For instance:
$$3a>1 \Rightarrow a>\frac{1}{3} \quad \mbox{ but } \quad -3a>1 \Rightarrow a >\frac{1}{-3} \Rightarrow a <\frac{1}{3}$$
In your case, you are dividing by $3K-4$. You do not know anything about its sign, since you do not know anything about $K$. Thus, any of the cases could be true. Or false.
If $K<4/3$, the first one is correct (but not the second one) since you are sure that your dividing quantity, $3K-4$, is positive and, hence, you do not need to change the sense of the inequality (note, again, what happens in my first example). It happens the same for your second case: it is true if $K>4/3$.
A: In the first approach, you divide by $4-3K$, and in the second by $3K-4$.
The direction of an inequality is preserved only when you multiply/divide by a positive quantity.
So which approach is "correct" depends on the sign of $4-3K$.

When you perform such transformations of inequalities, always discuss the sign of the multiplicand.
