flip or reverse inequality sign

Question

I have a basic math question.

If I have the following inequality: $$-a-b > -1$$ and I want to flip (or reverse) the sign. What is the correct way of the following? And why?

i) $a+b \le 1$
ii) $a+b < 1$

Many thanks! (:

Answer 1

The step is

$$-a-b > -1\iff (-a-b)(-1) \stackrel{reversed}{\color{red}<} (-1)(-1)\iff a+b<1$$

Let consider for a numerical example

$$1 > -1\iff 1(-1) < (-1)(-1)\iff -1<1$$

Note also that for $-a-b \ge -1$ the following holds

$$-a-b \ge -1\iff a+b\le1$$

Answer 2

Multiplying the given inequality by $-1$ we get $$a+b<1$$

Answer 3

$-a-b>-1$ $\implies$ $-a-b+(a+b+1)>-1+(a+b+1)$ $\implies$ $1>a+b$

Answer 4

Start with $-a-b>-1$.

Add $1+a+b$ to both sides to get $1>a+b$.

Which is the same as $a+b<1$. So ii).

That's why multiplying by a negative number reverses the inequality sign.

Just a comment: $a+b<1$ implies $a+b \leq 1$, but $a+b \leq 1$ does not imply $a+b<1$. Because properties of $<$ versus $\leq$. $$a+b \leq 1 \iff a+b<1 \text{ or } a+b=1.\\ a+b<1 \iff a+b\leq 1 \text{ and } a+b \neq 1.$$