I was struggling to get my head around proof by induction for inequalities when I came across a method described at The Math Forum (first answer). Here Dr. Ian goes ahead and compares the changes in the inequalities made by substitution ($n=k+1$), by assuming the original inequality to be equal.
In the original example $ \begin{align} \text{For }n=k&\quad (1+x)^k \geq (1+kx)\\ \text{For }n=k+1&\quad (1+x)^k\underbrace{(1+x)}_{\text{change}}\geq(1+kx)\underbrace{+\,x}_{\text{change}} \end{align} $ Now Dr. Ian claims that
there are two things left to do. The first is to show that (or explain the conditions under which) something multiplied by (1+x) is greater than the same thing plus x: $ \alpha * (1+x) >= \alpha + x $
I have found this method very appealing as it significantly simplifies the algebra - instead of expanding one side and forming a conjucture as to why it will be larger than the other, I only need to isolate $\alpha$ and show that it holds true when substituted back into the original inequality.
However my only concern is the assumption that $\alpha=(1+x)^k=(1+kx)$, since they are in fact not equal, but greater or equal. Is it fair to make this assumption nonetheless, making this method a valid strategy for proving inequalities by induction?