Graph linear inequalities in two variables

Section 6.7

#44 There is nothing strange in the circle being

the origin of any and every marvel. Aristotle

Concept

Up until this point we’ve discussed inequalities that involve only one dimension or one variable

Today we’re going to take our understanding of inequalities and apply it to two dimensions (variables)

First we will do a short review of lines and linear terms

Slope

Slope isA. An index of the angle of a lineB. A ratio of how much a line increases versus how much to moves right of leftC. A ratio of run to riseD. An index of movement in the x direction

Slope

What is the slope of the line that goes through the points (1,2) & (5,4)

4 3 22;(7, 2)x y

12

12

.

. 2

.

.1

A

B

C

D

Slope

What is the slope of the line that goes through the points (-4,-2) & (7,-8)

4 3 22;(7, 2)x y

1011

512

103

611

.

.

.

.

A

B

C

D

Slope

What is the slope of the line that goes through the points & 1 25 3,

53

157

. 3

.

.

.3

A

B

C

D

6 75 3,

Slope

What is the slope of the line that goes through the points (5,2) & (5,4)

35

15

.

.0

.

.3

A

B

C

D

Slope

What is the equation of the line that goes through the points (1,3) & (3,7)

5 12 2

5 192 2

. 2 1

. 2 13

.

.

A y x

B y x

C y x

D y x

Slope

What is the equation of the line that goes through the points (-3,4) & (-5,-12)

. 1

. 7

. 8 28

. 8 20

A y x

B y x

C y x

D y x

Slope

What is the equation of the vertical line that goes through the point (3,-5)

. 5

. 5

. 3

. 3

A y

B x

C y

D x

Slope

The equation of a line is y=3x-9. The slope of the line is increased by 2. What happens to the line?

A. The line has the same y-intercept, but now slopes downwardB. The line has the same y-intercept, but is now steeperC. The line has a different y-intercept, but now slopes downwardD. The line has a different y-intercept, but is now steeper

Slope

Assuming that the line starts at x=0, which line will reach y=50 first?

. 4 5

. 4 30

. 8

. 8 5

A y x

B y x

C y x

D y x

The big idea When we look at a line, we’re seeing the collection of

points that are solutions to a linear equality When looking at a linear inequality, instead of looking

at a set of points, we are seeing a defined space that indicates the infinite collection of points that satisfy the criteria

For example22 xy

Y

X22 xy

This means that any point that falls in the shaded

area is a viable solution to the inequality

Testing a point We can see this by testing out a point in the shaded

area For example

!

103

2123

2)6(23

Works

Y

X

22 xy

(-6,3)

It’s imperative that we remember that

the solution to these inequalities is an area as opposed

to a line

Process out of examples Our process for creating these graphs is not difficult,

but rather just an extension of our previous knowledge of graphing

Y

X

Graph the line via linear graphing methods

Draw a dashed line

for >,< otherwise a

solid line

Shade the appropriat

e area

Above for greater

than

Below for less than

Example Let’s do an example

Y

X

43 xy

Example How would we graph this one?

Y

X

6y

Example We would operate horizontal and vertical inequalities

the same as any other inequalityY

X

4x

ExamplesY

X

24

7y x

Example And another one

Y

X

32

1

1224

1242

xy

xy

yx

Example And another one

Y

X

9 3 12x y

Practical ExampleA party shop makes giftbags for birthday parties. They charge $4 per glowstick and $10 per T-shirt. Let x represent the number of glowsticks and y the number of T-shirts. The goal is to earn at least $500 from the sale of the bags• Write an inequality that describes the goal in terms of x & y• Graph the inequality• Give three possible combinations of pairs that will allow the shop to meet it’s goal

Y

X

Most Important Points What’s the most important thing that we can learn

from today? The solution to an inequality in two-dimensions is an area, as

opposed to a line We can graph the solutions to an equation by following our

normal processes for graphing lines and then shading the appropriate area

Homework

6.7 you will have two days1, 2-32, 47-50, 53-57