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.

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

An equation like

y = 2x + 1

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

y > 2x + 1

is a whole region: every point lying above that line. So a graphical inequality splits the plane into "satisfies it" and "doesn't", and the boundary is 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.

(When in doubt, test one point — e.g. the origin — and shade the side that works.)

Dashed or solid boundary. A strict inequality (

<

>

) does not include the boundary, so draw it dashed. An inclusive one (

\le

\ge

) includes it, so draw it solid.

Several inequalities at once. A region defined by more than one inequality is the overlap — the set of points satisfying all of them simultaneously. So sketch each boundary, shade each region, and the answer is where the shadings coincide. Its corners are where boundaries cross, found by solving the boundary equations simultaneously.

Here are the boundaries

y = x^2

and

y = 4

; the region

x^2 \le y \le 4

(with

x \ge 0

) is the slice trapped between them in the first quadrant, with corners

(0,0)

(0,4)

and

(2,4)