Quadratic Inequalities in Two Variables

Solve and graph quadratic inequalities in two variables. Math 20-1 Alberta.

Lesson 2.7 of Linear Functions & Systems of Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Understanding the Solution Region. Learn what quadratic inequalities in two variables look like and how to test whether any given point belongs to the solution region.
• Graphing the Solution Region. Follow a five-step process to graph any quadratic inequality: isolate y, find key features of the parabola, choose solid or dashed, shade the correct region, and verify with a test point.
• Writing an Inequality from a Graph. Work backwards from a graph — use vertex form to find the equation of the parabola, then read the curve type and shading to choose the correct inequality symbol.

Inside the lesson — a free preview

What Does a Quadratic Inequality Look Like?

Definition — Quadratic Inequality in Two Variables. A quadratic inequality in two variables has one of these four forms:

y < ax^2 + bx + c

y > ax^2 + bx + c

y ≤ ax^2 + bx + c

y ≥ ax^2 + bx + c

Solving the inequality means determining a solution region — the set of all (x, y) pairs that make the statement true. This region lies above or below the parabola, and may or may not include the points on the parabola itself.

The Solution Is a Shaded Graph. Unlike a quadratic equation (which gives a curve), a quadratic inequality gives a shaded region. Every point in the shaded region satisfies the inequality. Every point outside does not.

Checking Whether a Point Is a Solution

The Test-Point Method. Before graphing, it helps to verify individual points. To test whether an ordered pair (x, y) is a solution:

Step 1: Substitute the x and y values of the point into the inequality.

Step 2: Simplify the right side.

Step 3: Decide whether the resulting statement is TRUE or FALSE. A true statement means the point IS in the solution region.

Worked example: Which Points Are Solutions to y ≥ x^2 - 3x - 4?

Test each of these points: (0, 0), (2, -2), (7, 1), (-5, -2), (-3, 0), (1, 4).

Test (0, 0). 0 ≥ (0)^2 - 3(0) - 4

0 ≥ -4 TRUE

Test (2, -2). -2 ≥ (2)^2 - 3(2) - 4

-2 ≥ 4 - 6 - 4

-2 ≥ -6 TRUE

Test (7, 1). 1 ≥ (7)^2 - 3(7) - 4

1 ≥ 49 - 21 - 4

1 ≥ 24 FALSE

Test (-5, -2). -2 ≥ (-5)^2 - 3(-5) - 4

-2 ≥ 25 + 15 - 4

-2 ≥ 36 FALSE

Test (-3, 0). 0 ≥ (-3)^2 - 3(-3) - 4

0 ≥ 9 + 9 - 4

0 ≥ 14 FALSE

Test (1, 4). 4 ≥ (1)^2 - 3(1) - 4

4 ≥ 1 - 3 - 4

4 ≥ -6 TRUE

Summary. Solutions: (0, 0), (2, -2), and (1, 4) all make the inequality true.

Non-solutions: (7, 1), (-5, -2), and (-3, 0) make the inequality false — they lie outside the solution region.

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