Solve inequality equation:
x^2-5x-6<0
An algebraic solution is
$\displaystyle x^2-5x-6<0\Rightarrow\ (x-6)(x+1)<0$
This is negative if the product of the factors gives (-)(+)
If $\displaystyle x<-1,$ both factors are negative
and if $\displaystyle x>6,$ both factors are positive, hence in both cases the product will be positive.
Therefore, one can find from this the valid range of x.
Alternatively
$\displaystyle x^2-5x-6<0\Rightarrow\ x^2-5x<6\Rightarrow\ x(x-5)<6$
$\displaystyle x>6\Rightarrow\ (+)(+)>6$
$\displaystyle x<-1\Rightarrow\ (-)(-)>6$
Again, the range of x can be deduced.