Question

How do you graph linear inequalities in two variables?

Answer 1

The best way (when possible!) is to express the inequality bringing all the terms involing a variable on the left, and all the terms involving the other variable on the right. In the end, you'll have an inequality of the form `y\le f(x)` (or `y\ge f(x)`), and this is easy to graph, because if you can draw the graph of `f(x)`, then you'll have that `y\le f(x)` represents all the area under the function `f`, and `y\ge f(x)`, of course, the area over the function.

For example, consider the inequality
`yx^2<-y+3x`
In this form, it would be very hard to say which points satisfy the inequality, but with some manipulations we obtain
`yx^2+y<3x`
`y(x^2+1)<3x`
`y<\frac{3x}{x^2+1}`

Now, the graph of `\frac{3x}{x^2+1}` is easy to draw, and the inequality is solved considering all the area below the graph, as showed:

graph{y<3x/{x^2+1} [-10, 10, -5, 5]}

Note that if you have `y<f(x)` the graph of the function `f` is not included, since it represents the points for which `y=f(x)`.