Prove modular inequalities $ab + ac\le a(b+ac)$ and $(a+b)(a+c)\ge a+b(a+c)$

Prove modular inequalities $ab + ac\le a(b+ac)$ and $(a+b)(a+c)\ge a+b(a+c)$

How to prove
$$(a\cdot b)+(a\cdot c)\le a\cdot\big(b+(a\cdot c)\big)$$
and
$$(a+b)\cdot(a+c)\ge a+\big(b\cdot(a+c)\big)\;?$$
I have tried this. Using distributive property, I think we can get
$$a+(b\cdot c) \le (a+b)\cdot(a+c)$$
and
$$a\cdot(b+c) \ge (a\cdot b) + (a\cdot c)\;.$$
Now what should I do?