The idea

Inequalities on graphs and regions

Reading an inequality as a region of the plane — which side of a curve to shade, when the boundary is dashed or solid, and how overlapping inequalities define a region.

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Pure · Algebra & functions

Inequalities on graphs and regions

Reading an inequality as a region of the plane — which side of a curve to shade, when the boundary is dashed or solid, and how overlapping inequalities define a region.

Why it works

A line is thin; an inequality is a region

Swap one symbol and the answer changes dimension. An equation like

y = 2x + 1

is a line — a one-dimensional thread of points. The inequality

y > 2x + 1

is a whole region: every point lying above that line, a two-dimensional expanse. So a graphical inequality splits the plane into "satisfies it" and "doesn't", and the boundary between the two is always the matching equation.

Which side?

Compare

y

with the curve's value:

$y > f(x)$ is the region above the curve $y = f(x)$ ,
$y < f(x)$ is the region below it.

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