inequality - Why is Mathematical Induction used to prove solvable inequalities?

As a first year undergrad student I've seen problems where solvable inequalities need to be proven to hold in a specific domain using Mathematical Induction.

My question is, if the inequalities are solvable, i.e. where the bounds where the inequality holds true, it's domain where it holds, can be solved for, why is there a need for induction to prove the inequality holds true?

For example this was one of the problems where I had to prove the inequality held true throughout the domain.

Prove using Mathematical Induction: $$n^2 \geq 2n +3, \forall n\in \mathbb{Z} \geq3$$

But I could far easier solve the inequality for the domain where the inequality is true. As shown below.
$$ Let, P(n) = n^2 \geq 2n+3$$
$$ Let, D = \{n \in \mathbb{Z} | n \geq 3 \}$$

Required to prove:
$$P(n) \forall n \in D$$
Proof :
$$ n^2 -2n -3 \geq 0 $$
$$ (n-3)(n+1) \geq 0 $$
$$ (n \geq 3\wedge n \geq -1) \vee (n \leq 3 \wedge n \leq -1)$$
$$ (n \geq 3) \vee (n \leq -1)$$
$$ n \in \mathbb{R} (-\infty, 1] \cup [3, \infty)$$

Since $$\mathbb{Z} \subset \mathbb{R}$$

we have proven P(n) for all elements in D. $$Q.E.D$$

First off is what I've done above Mathematically, correct, and furthermore is it formally correct? I've written what I set out to prove in symbolic form, which is why some of the proof may seem a bit redundant.

Secondly why would I need to use Mathematical Induction for cases like these where the domains in which the inequality holds true is easily solvable, what is the upside of using Mathematical Induction in this case?

Thirdly I was told that Mathematical Induction is great to use for natural numbers and integers, but that it fails for real numbers. If that is correct wouldn't that favor using the direct solving of the inequality root as I did above, as solving the inequality would solve for the domain in which it is true for up to real and complex numbers if I'm not mistaken.

Finally if you have spotted any errors in my reasoning or proof writing techniques/skills, please feel free to criticize and inform me of it, I'm majoring in Math, so don't hold back on me.

Answer

As many users commented above, these sorts of fairly trivial questions are mainly given to students to increase their familiarity with Mathematical Induction.