Inequalities: Do these properties hold?

Question

I was curious if the following inequalities hold.

Inequality #1

Assume $A, B, C, D$ are real numbers.

Given:

$A\le C$

$B\le D$

Does $A+B\le C+D$ hold?

Inequality #2

Assume $A, B, C, D$ are real numbers, and $A, C\ge 0$.

Given:

$A\le C$

$B\le D$

Does $AB\le CD$ hold?

Answer 1

We have $\color{red}{A \leq C}$ and $\color{blue}{B \leq D}$ so \begin{eqnarray*} \color{red}{A}+B \leq \color{red}{C}+\color{blue}{B} \leq C +\color{blue}{D}. \end{eqnarray*} The other inequality follows similarly provided the values are positive.

Answer 2

For inequality #1)

Follow from given information, $A+B\le A+D\le C+D$, using transitive property and addition property.

Simialaly,If $A,B,C,D\ge 0$, inequality#2 holds.

However, only assuming $A,C\ge 0$, IT IS NOT TRUE. Consider the case $(A,B,C,D)=(0,-2,1,-1)$

Answer 3

The first inequality is absolutely correct.

The second inequality however:

If $0 \le B \le D$ then the inequality is correct.

If $B \le D \le 0$ then $-B, -D \ge 0$ and $-D\le-B$, combine with $0 \le A \le C$, only the inequality $0 \le -AD \le -BC$ or $AD \ge BC$ is absolutely true, we can't infer anything from it to prove the comparison between $AB$ and $CD$.

If $B \le 0 \le D$, because $0 \le A \le C$, we will have $AB \le 0 \le CD$, the inequality is true.