# Inequality $((4000 - x)^2 - (3200 - x)^2) > ((2100 - y)^2 - (4000 - y)^2)$

I've been working with a distance formula and trying to come up with an efficient way of figuring out the range of $$y$$. Here's the formula:

$$((4000 - x)^2 - (3200 - x)^2) > ((2100 - y)^2 - (4000 - y)^2)$$

Basically, I will always know the $$x$$ value beforehand. I want an efficient way that when given $$x$$, will produce the range of $$y$$ that will make this inequality true.

This is for a program I'm coding, where simply going through all possible $$y$$ values is way too slow.

When I go on a site like Desmos, it plots it so quickly! I'd appreciate any help. Thanks!

So you have $$C > ((2100 - y)^2 - (4000 - y)^2)$$
Thus $$C > (2100^2 - 4000^2 + 3800y)$$
$$C + 4000^2 - 2100^2 > 3800y$$
$$y < \frac{C + 11590000}{3800}$$ are values of y which satisify the inequality

• This is great, the steps were a good help too. Thanks for the solution! I don't know where my head was. Jul 7, 2021 at 7:57

Expanding both sides, we obtain

$$1600(3600 - x) > 3800(y - 3050),$$

which simplifies to

$$y < -\frac{2}{19}(4x - 43375).$$

• Thanks a lot, seeing this now makes me wonder where my head was. Greatly appreciated. Jul 7, 2021 at 7:54