1. ## Solving algebraic inequalities

Hi, I'm having trouble solving this inequality:

'Let $\displaystyle p$ and $\displaystyle q$ be positive real numbers'. Prove that:

$\displaystyle (p + 2)(q + 2)(p + 1) \geq 16pq$

Thanks, BL

2. Originally Posted by BG5965
Hi, I'm having trouble solving this inequality:

'Let $\displaystyle p$ and $\displaystyle q$ be positive real numbers'. Prove that:

$\displaystyle (p + 2)(q + 2)(p + 1) \geq 16pq$
This is not true for example if p = q = 6. Then the left side is 448 and the right side is 576.

3. Maybe the inequality should be

$\displaystyle (p+2)(q+2)(p+q)\geq 16pq$.

That can be proved as follows.

$\displaystyle (x - y)^2 \geq 0$
$\displaystyle x^2 + y^2 - 2xy \geq 0,$
$\displaystyle x^2 + y^2 \geq 2xy.$

Now let $\displaystyle x = \sqrt{p}$ and $\displaystyle y = \sqrt{q}$, then

$\displaystyle p + q \geq 2\sqrt{pq}.$

Similarly $\displaystyle q + r \geq 2\sqrt{qr}$ and $\displaystyle r + p \geq 2\sqrt{rp}$.

Multiplying these three inequalities and putting $\displaystyle r = 2$, leads to the stated result.