Question

How do you graph the inequality `–3x – 4y<=12`?

Answer 1

To graph this kind of inequality, the best thing is to manipulate them and obtain a relation of the form `y\le f(x)` (or `y \ge f(x)`).

In fact, we know that `y=f(x)` is exactly the graph of `f`, and so `y\le f(x)` (or `y \ge f(x)`) represents all the portion of the plan below (or above) the graph of `f`.

Let's do those manipulations: starting from
`-3x-4y\le 12`,
adding `4y` at both sides we get
`-3x \le 4y+12`.
Subtracting 12 at both sides, we have
`-3x-12\le 4y`.
Dividing both sides by `4` we finally have
`-3/4 x - 3 \le y`
which can of course be read as
`y \ge -3/4 x - 3`

We thus have `f(x)=-3/4 x - 3`, which is a line and so it's very easy to graph. Once graphed, you need to consider all the portion of plan above the line to solve your inequality.

Here's the graph: graph{-3x-4y \le 12 [-10, 10, -5, 5]}