Linear Inequalities in Two Variables

Objectives: Solve and graph a linear inequality in two variables.

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A linear inequality in two variables, x and y, is any inequality that can be written in one of the forms below, where A ≠ 0 and B ≠ 0.

Ax + By ≥ C Ax + By > C Ax + By ≤ C Ax + By < C

A solution of a linear inequality in two

variables, x and y, is an ordered pair

(x, y) that satisfies the inequality. The

solution to a linear inequality is a region of the

coordinate plane and is called a half-plane

bounded by a boundary line.

Graphing Linear Inequalities

1. Given a linear inequality in two variables, graph its related linear equation.

• For inequalities involving ≤ or ≥, use a solid boundary line.

• For inequalities involving < or >, use a dashed boundary line.

2. Shade the appropriate region.

• For inequalities in the form of y ≤ mx + b or y < mx + b, shade below the boundary line.

• For inequalities of the form y ≥ mx + b or y > mx + b, shade above the boundary line.

• For inequalities in the form x ≤ c or x < c, shade to the left of the boundary line.

• For inequalities in the form x ≥ c or x > c, shade to the right of the boundary line.

Ex 1. Graph each linear inequality.

• a. y < x + 2

b. y ≥ -2x + 3

* c. y > -2x - 2

Dotted Line

d. y ≥ 2x + 5

e. -2x –3y ≤ 3

f. 3x – 4y ≥ 4

-4y≥-3x + 4

y ≤ ¾ x - 1

g. -5x – 2y > 4

-2y > 5x + 4

y < -5/2 x - 2

Dotted Line

Ex 3. Graph each linear inequality. x is a vertical line

and

y is a horizontal line

a. x > -2

b. y ≤ -1

c. x ≤ -2

d. y > -1 Dotted Line