3.3 Linear Inequalities in Two Variables Objectives: Solve and graph a linear inequality in two variables. Use a linear inequality in two variables to solve real-world problems. Standard: K. Apply an appropriate technique to graph a linear inequality.

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 Ax + By ≤ C Ax + By < C

A solution of a linear inequality is the value of the variables (x, y) that make the inequality a true statement. A solution of a linear inequality is the value of the variables (x, y) that make the inequality a true statement. When graphed on a coordinate plane the solution to a linear inequality is a region called a half-plane and is bordered by a boundary line. When graphed on a coordinate plane the solution to a linear inequality is a region called a half-plane and is bordered by a boundary line.

Graphing Linear Inequalities 1. When graphing a linear inequality with two variables, graph the border line as if it were a linear equation. For inequalities involving ≤ or ≥, use a solid boundary line. For inequalities involving ≤ or ≥, use a solid boundary line. For inequalities involving, use a dashed boundary line. For inequalities involving, use a dashed boundary line.

2. After graphing the border line, shade the appropriate region. For inequalities in the form of y ≤ mx + b or For inequalities in the form of y ≤ mx + b or y < mx + b, shade below the boundary line. y < mx + b, shade below the boundary line. For inequalities of the form y ≥ mx + b or For inequalities of the form y ≥ mx + b or y > mx + b, shade above the boundary line. y > mx + b, shade above the boundary line. For inequalities in the form x ≤ c or x < c (vertical lines), shade to the left of the boundary line. For inequalities in the form x ≤ c or x < c (vertical lines), shade to the left of the boundary line. For inequalities in the form x ≥ c or x > c (vertical lines), shade to the right of the boundary line. For inequalities in the form x ≥ c or x > c (vertical lines), shade to the right of the boundary line.

Ex 1. Graph each linear inequality. a. y < x + 2 a. y < x + 2

b. y ≥ -2x + 3

* c. y > -2x - 2

d. y ≥ 2x + 5

e. -2x –3y ≤ 3

f. 3x – 4y ≥ 4 -4y≥-3x + 4 y ≤ ¾ x - 1

g. -5x – 2y > 4 g. -5x – 2y > 4 -2y > 5x + 4 y < -5/2 x - 2 Dotted Line

Vertical line: x > a constant x a constant Ex 3. Graph each linear inequality. Horizontal line: y > a constant y < a constant y > a constant

a. x > -2

b. y ≤ -1

c. x ≤ -2 c. x ≤ -2

d. y > -1

Writing Activities