Solving &

GraphingInequalities

Half - Plane

Objective- To graph inequalities on the coordinate plane.

Recall…Graph n < 3 on a number line.

-3 -2 -1 0 1 2 3 4

Graph y > 3 on the coordinate plane.

x

y

Graph x -2 on the coordinate plane.

x

y

How to graph a line

• Write the inequality as an equation• E.g. y ≥ - 3x + 2 y = -3x + 2• Make a table of x,y values• x y = -3x+2• 0 -3(0) + 2 = 2 (0,2)• 1 -3(1) +2 = -1 (1, -1)• 2 -3(2) +2 = -4 (2, -4)• Graph these points and draw a line.

Graph y -3x + 2 on the coordinate plane.

x

y

Boundary Line

y = -3x + 2

a = -3

b = 2

31

Test a point not on the linetest (0,0)

0 -3(0) + 2Not true!

Graph y -3x + 2 on the coordinate plane.

x

y

Instead of testing a point

If in y = ax + b form...

Shade up

Shadedown

Solid line

Dashed line

> <

Graph on the coordinate plane.3x - 4y > 12-3x -3x-4y > -3x + 12-4 -4y < x - 3

34

a =

b = -3

34

Boundary Linex

y

ProblemIf you have less than $5.00 in nickels and dimes, find an inequality and sketch a graph to describe how many of each coin you have.

Let n = # of nickelsLet d = # of dimes

0.05 n + 0.10 d < 5.00 or 5 n + 10 d < 500

5n + 10d < 500n d

0 50

100 0

0 10 20 30 40 50 60 70 80 90 100n

d

60

50

40

30

20

10

0

Find the Vertex(es) or Vertices

• What were the (x,y) coordinates for the triangle of constraints we just made?

Optimization

• Optimization is when we use a polygon of constraints (the shape we made with the inequalities) to find out the maximum or minimum value of another equation.

• We need the vertices: (0,50) (100,0) (0,0)• We substitute them into another equation to

and calculate the value.

Optimization

x y Optimization Equation

P = 3x – 2 y0 0 3(0) -2(0)

= 00 50 3(0) -2(50)

= -100 Min100 0 3(100) -2(0)

= 300 Max

Check-up

AssignmentPage 43, Q. 3,4,6

**Will need graph paper