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?

I guess you have the precondition $a,b,c\geq 1$.
Then
$$ab + ac = a(b+c) = a(b+1\cdot c)\overset{a\geq 1}{\leq} a(b + ac)$$
and
$$(a+b)(a+c) = a\underbrace{(a+c)}_{\geq 1} + b(a+c) \geq a + b(a+c).$$
