Linear Inequalities in Two Variables

Graph and solve linear inequalities in two variables. Math 20-1 Alberta Grade 11.

Lesson 2.6 of Linear Functions & Systems of Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Graphing Linear Equations (Quick Review). Review slope-intercept form and general form, and practice sketching four types of lines including horizontal and vertical lines.
• Linear Inequalities in Two Variables. Learn how to draw a boundary line, choose solid or dotted, shade the correct half-plane, and verify with a test point — then go backwards from a graph to write the inequality.

Inside the lesson — a free preview

Two Common Forms

Two Ways to Write a Linear Equation. Every linear equation can be expressed in either slope-intercept form or general form. The form you choose determines the most efficient graphing strategy.

Which Form Should You Use?. Either method works for any linear equation. Use whichever feels easier — slope-intercept is fastest if y is already isolated. The x-intercept and y-intercept method is convenient when working with general form since setting one variable to 0 quickly gives two points.

Worked example: Sketch Each Line

Part a) — y = (1)/(2)x - 3. Already in slope-intercept form.

Step 1: Identify: y-intercept = -3, slope = (1)/(2).

Step 2: Plot (0, -3). From there, rise 1 and run 2 to reach the second point (2, -2).

Step 3: Draw a straight line through both points.

Part b) — 5x - 2y + 10 = 0. In general form. First convert to slope-intercept form.

Step 1 — Isolate y:

5x - 2y + 10 = 0

-2y = -5x - 10

y = (5)/(2)x + 5

Step 2 — Graph: y-intercept = 5, slope = (5)/(2). Plot (0, 5), then rise 5 and run 2 to (2, 10).

Part c) — y - 2 = 0. Simplifies to y = 2, a horizontal line. Every point on this line has a y-value of 2, regardless of x.

Key rule: When there is no x in the equation, the result is a horizontal line.

Part d) — x + 5 = 0. Simplifies to x = -5, a vertical line. Every point on this line has an x-value of -5, regardless of y.

Key rule: When there is no y in the equation, the result is a vertical line.

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