How do I solve x + 1 = 3 without a calculator?

Treat the equation like a balanced scale. Whatever you do to one side, you must do to the other side. Start with: $$x+1=3.$$ The $+1$ is attached to $x$, so undo it by subtracting $1$ from both sides: $$x+1-1=3-1.$$ That simplifies to: $$x=2.$$ Now check it by putting $2$ back into the original equation: $$2+1=3.$$ That is true, so the solution is: $$x=2.$$ The reason this is allowed is that subtracting the same number from both sides keeps the equality true. You are not changing the equation unfairly; you are rewriting it in an equivalent form where $x$ is alone.