3.3 Linear Inequalities 3.3 Linear Inequalities in Two Variablesin 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: 2.8.11.K. Apply an appropriate technique to graph a linear inequality.

A A linear inequality in two linear inequality in two variablesvariables, x and y, is any , x and y, is any

inequality that can be written in inequality that can be written in one of the forms below, where A ≠ one of the forms below, where A ≠

0 and B ≠ 0.0 and B ≠ 0.

AAxx + B + By y ≥ C A≥ C Axx + By > C + By > C A Axx + B + Byy ≤ C A ≤ C Axx + B + Byy

< C< C

A solution of a linear inequality in two A solution of a linear inequality in two

variables, variables, xx and and yy, is an ordered pair , is an ordered pair

((xx, , yy) that satisfies the inequality. The ) that satisfies the inequality. The

solution to a linear inequality is a region of solution to a linear inequality is a region of the the

coordinate plane and is called a coordinate plane and is called a half-half-planeplane

bounded by a bounded by a boundary lineboundary line..

Graphing Linear Inequalities Graphing Linear Inequalities

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

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

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

2. Shade the appropriate region.2. Shade the appropriate region.

For inequalities in the form of For inequalities in the form of yy ≤ ≤ mxmx + + bb or or yy < < mxmx + + bb, shade below the boundary line., shade below the boundary line.

For inequalities of the form For inequalities of the form yy ≥ ≥ mxmx + + b b or or y y > > mxmx + + bb, shade above the boundary line., shade above the boundary line.

For inequalities in the form For inequalities in the form xx ≤ ≤ cc or or xx < < cc, , shade to the left of the boundary line.shade to the left of the boundary line.

For inequalities in the form For inequalities in the form xx ≥ ≥ cc or or xx > > cc, , shade to the right of the boundary line.shade to the right of the boundary line.

Ex 1. Graph each linear Ex 1. Graph each linear inequality.inequality.

a. a. yy < < xx + 2 + 2

b. b. yy ≥ -2 ≥ -2xx + 3 + 3

** c. c. yy > -2 > -2xx - 2 - 2

Dotted Line

d. d. yy ≥ 2 ≥ 2xx + 5 + 5

e. -2e. -2xx –3 –3yy ≤ 3 ≤ 3

f. 3x – 4y ≥ 4f. 3x – 4y ≥ 4

-4y≥-3x + 4

y ≤ ¾ x - 1

g. -5x – 2y > 4g. -5x – 2y > 4 -2y > 5x + 4

y < -5/2 x - 2

Dotted Line

Ex 3. Graph each linear Ex 3. Graph each linear inequality. inequality.

x is a x is a vertical linevertical line

and and

y is a y is a horizontal linehorizontal line

a. a. xx > -2 > -2

b. b. yy ≤ -1 ≤ -1

c. c. xx ≤ -2 ≤ -2

d. y > -1 Dotted Line

Writing ActivitiesWriting Activities