Graph solution regions of linear and quadratic inequalities in two variables

Alberta Grade 11 Mathematics

Relations and Functions

Test the region, then decide whether the boundary belongs

An inequality in x and y describes an entire region, not just a curve. First graph the related equality as a boundary; then decide which side contains solutions. In Mathematics 20-1, a strict inequality makes the boundary dashed, whereas an inequality with equality includes it. The diagram below computes the curve and shaded region from the same coefficients used in the worked example.

What you will learn

Graph a quadratic boundary.
Use a test point to choose the solution side.
Distinguish dashed and solid boundaries.

Draw the boundary equation

For

y>x^2-4

, temporarily replace > with = to get the parabola

y=x^2-4

. Its vertex is

(0,-4)

and its x-intercepts are −2 and 2.

Because the original sign is strict, equality is not a solution. Draw the boundary as a dashed curve. If the sign had been ≥ instead, draw a solid boundary.

y=x^2-4

Find the correct side

A test point not on the boundary tells you which side to shade. At

(0,0)

, the statement

0>0^2-4

is true, so shade the region containing the origin, above the parabola.

A point below the boundary, such as

(0,-5)

, fails the inequality because −5 > −4 is false. This second check helps catch accidental shading on the wrong side. The picture shows only a finite window of the infinite solution region.

0>0^2-4

Worked example

Graph the solution region

Graph

y>x^2-4

and decide whether

(0,0)

,

(0,-4)

, and

(0,-5)

are solutions.

Solution region for y > x² − 4

Green shading is above the dashed parabola. The boundary itself is excluded; the region continues outside this window.

Graph the boundary
The equality has vertex (0, −4) and roots −2 and 2. Use a dashed line because the inequality is strict.
$y=x^2-4$
Choose a side
The origin passes the test, so shade above the curve.
(0,0): 0>-4 $\text{true}$
Check the three points
(0, −4) lies exactly on the excluded boundary; (0, −5) lies below it.
(0,0) $\text{ yes}$ ; (0,-4) $\text{ no}$ ; (0,-5) $\text{ no}$

Answer: Shade the region above

y=x^2-4

with a dashed boundary. Only

(0,0)

of the three test points is included.

Check: Substituting each coordinate into the original strict inequality gives true, false, false.

Common mistakes and how to avoid them

Drawing a solid curve for >.

Correction: Strict > and < exclude equality, so the boundary is dashed.

Shading below without testing.

Correction: Substitute a point off the curve, such as the origin, before shading.

Lesson summary

The related equality is the boundary.
A test point chooses the solution side.
Strict inequalities exclude the boundary.

Check your understanding

Question 1

For

y\geq x^2

, which boundary is correct?

Solid parabola
Dashed parabola
No boundary

Show answer and explanation

Solid parabola

The equality portion of ≥ includes points on the parabola.

Key terms

Boundary: The curve where the two sides of an inequality are equal.
Solution region: All coordinate pairs that make the inequality true.

Continue through Math 20-1

View the complete Math 20-1 Alberta Grade 11 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-1), expectation RF7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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RF7 · Graph solution regions of linear and quadratic inequalities in two variables