Manipulating Inequalities

In algorithm analysis, you often need to prove that one expression is bounded by another. This requires manipulating inequalities.

For example, to show $2n + 3 \leq 5n$ for $n \geq 1$, subtract $2n$ from both sides: $3 \leq 3n$, then divide by $3$: $1 \leq n$. This is true when $n \geq 1$.

These steps justify why you can write $O(2n + 3) = O(n)$ for large $n$. The algebra proves the simplification.