How do you graph linear inequalities in two variables?

1 Answer
KillerBunny
Feb 1, 2015

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)#.

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