# Math Help - Solving algebraic inequalities

1. ## Solving algebraic inequalities

Hi, I'm having trouble solving this inequality:

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

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

Thanks, BL

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

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

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

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

That can be proved as follows.

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

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

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

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

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