How do you graph linear inequalities in two variables?

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How do you graph linear inequalities in two variables?

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Answer 1 · 2015-02-01T12:26:16

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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How do you graph linear inequalities in two variables?

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