Do Now 1/5/12Do Now 1/5/12

In your notebook, explain how to graph the In your notebook, explain how to graph the following line; y = 4x – 3 following line; y = 4x – 3

ObjectiveObjective

SWBAT graph linear equations and SWBAT graph linear equations and linear inequalities in two variables linear inequalities in two variables

Section 6.7 “Graph Linear Section 6.7 “Graph Linear Inequalities”Inequalities”

Linear Inequalities-Linear Inequalities- the result of replacing the = signthe result of replacing the = signin a linear equation with an inequality sign. in a linear equation with an inequality sign.

Linear InequalitiesLinear InequalitiesAn example of a linear inequality in An example of a linear inequality in

two variables is two variables is x - 3y x - 3y ≤≤ 6. 6. The solution of The solution of an inequality in two variables, an inequality in two variables, x x andand y y, is an , is an ordered pair ordered pair (x, y)(x, y) that produces a true that produces a true statement when substituted into the statement when substituted into the inequality. inequality.

Which ordered pair is NOT a solution of x - 3y Which ordered pair is NOT a solution of x - 3y ≤ 6≤ 6??

A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)

x - 3y ≤ 6x - 3y ≤ 6

Substitute each point into the inequality. If the statement is true then it is a solution.

(0) – 3(0) ≤ 6(0) – 3(0) ≤ 6True, therefore(0,0) is a solution.

The graph of an inequality in two variables The graph of an inequality in two variables is the set of points that represent all is the set of points that represent all solutions of the inequality. solutions of the inequality.

The The BOUNDARY LINEBOUNDARY LINE of a linear of a linear inequality divides the coordinate plane into inequality divides the coordinate plane into two two HALF-PLANESHALF-PLANES. Only one half-plane . Only one half-plane contains the points that represent the contains the points that represent the solutions to the inequality. solutions to the inequality.

Graph an Inequality Graph an Inequality in Two Variables in Two Variables

Graphing Linear Inequalities Graphing Linear Inequalities

Graphing Graphing Boundary LinesBoundary Lines::

– Use a dashed line for < or >.Use a dashed line for < or >.

– Use a solid line for Use a solid line for ≤ or ≥.≤ or ≥.

Graph an InequalityGraph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality y > 4x - 3 y > 4x - 3..

STEPSTEP 22

Test (0,0) in the original inequality.

STEPSTEP 33

Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

34 xy 34 xy

3)0(40

True

Graph an InequalityGraph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality x + 2y ≤ 0..

STEPSTEP 22

Test (1,0) in the original inequality.

STEPSTEP 33Shade the half-plane that does not contain the point (1,0), because (1,0) is not a solution to the inequality.

02 yx 02 yx

0)0(21

False

Graph an InequalityGraph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality x + 3y ≥ -1..

STEPSTEP 22

Test (1,0) in the original inequality.

STEPSTEP 33Shade the half-plane that contains the point (1,0), because (1,0) is a solution to the inequality. 13 yx 13 yx

1)0(31

True

Graph an InequalityGraph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality y ≥ -3..

STEPSTEP 22

Test (2,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

STEPSTEP 33Shade the half-plane that contains the point (2,0), because (2,0) is a solution to the inequality. 3y

3y

3)0(

True

Graph an InequalityGraph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality x x ≤ -1..

STEPSTEP 22

Test (3,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

STEPSTEP 33Shade the half-plane that does not contain the point (3,0), because (3,0) is not a solution to the inequality.

1x

1x

1)0(

False

ChallengeChallenge

“Can You Write and Graph the Mystery “Can You Write and Graph the Mystery Inequality???”Inequality???”

The points (2,5) and (-3, -5) lie on the The points (2,5) and (-3, -5) lie on the boundary line. The points (6,5) and (-2, -3) boundary line. The points (6,5) and (-2, -3) are solutions of the inequality. are solutions of the inequality.

y y ≤ 2x + 1≤ 2x + 1

ABSOLUTE VALUE FUNCTION-ABSOLUTE VALUE FUNCTION-

Graph Absolute Value FunctionsGraph Absolute Value FunctionsExtension Activity 6.5Extension Activity 6.5

xx f(x)=|x|f(x)=|x|

-5-5 55

-2-2 22

00 00

11 11

33 33

f(x) = |x|

f(x) = |x|f(x) = |x|g(x) = |x -3|

xx g(x)=|x-3|g(x)=|x-3|

-3-3 66

00 33

33 00

44 11

66 33

g(x) = |x - 3|g(x) = |x - 3|