Bell Work… Graph the following equations. 1. 2. 6 4 2 2 14365 1 3 5 6 4 2 2 14365 1 3 5

Bell Work…Graph the following equations.

1. 2. x 3y− 3−=2x 4y+ 8=

66

44

22

22

11 4433 6655

11

33

55

66

44

22

22

11 4433 6655

11

33

55

Solving Systems of Equations

Graphing Linear Inequalities

Objectives

• How do we graph an inequality • Define a boundary line• Graphing a boundary line• Define the solution for a system of

inequalities• Find the solution of a system of

inequalities

What is the solution of an inequality

• Solution of an inequality are all the ordered pairs (points) that make the inequality true.

Graphing Inequalities

Consider the inequality

y ≥ x

66

44

22

22

11 4433 6655

11

33

55

y = x Graph

Boundary line

REMEMBER: Solution are all the ordered pairs (points) thatmake the inequality true.



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

1. Pick two points from each side of the graph

(4,1)

(1,3)



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)

2. Check points if they make inequality true.

(1,3) y ≥ xsubstitute into



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)



3 ≥ 1



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)



3 ≥ 1



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)





y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)



1 ≥ 4



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)



1 ≥ 4 X

X



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(4,1)

(1,3)

3. Shade the side where the correct point lies.

X



y ≥ x

66

44

22

22

11 4433 6655

11

33

55

(1,3)

3. Shade the side where the correct point lies.



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22

x - 2y = 4 Graph

y = x - 212



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22

x - 2y = 4 Graph

¡¡TEST POINTS !!

(0,1)

(6,0)

y = x - 212



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22(0,1)

(6,0)

(0,1)substitute into

x - 2y ≤ 4



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22(0,1)

(6,0)


x - 2y ≤ 4

0 - 2(1) ≤ 4

-2 ≤ 4



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22(0,1)

(6,0)


x - 2y ≤ 4



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22(0,1)

(6,0)


x - 2y ≤ 4

6 - 2(0) ≤ 4

6 ≤ 4 X

X



x - 2y ≤ 4

33

11

22-1-1 11 4433 6655

-2-2

22(0,1)

(6,0)

X

¡¡ SHADE CORRECT REGION !!

Examples

66

44

22

22

11 4433 6655

11

33

55

3y - 2x ≥ 91.

y = x + 3 23

GRAPH

Examples

66

44

22

22

11 4433 6655

11

33

55

3y - 2x ≥ 91.

y = x + 3 23

GRAPH

TEST!!

(0, 5)(0,5)

(3,0)

X

3(5) - 2(0) ≥ 9

15 - 0 ≥ 9

Examples2.

x - 3y > -3

y = x + 1 13

66

44

22

22

11 4433 6655

11

33

55

Graph

TEST!!

(0, 5) (0,5)

0 - 3(5) > -3

0 - 15 > -3

X

X

Solving a system of Inequalities

Consider the system

x + y ≥ -1

-2x + y < 233

11

22

-1-1

11 33

22

-2-2-3-3 -1-1


Consider the system

x + y ≥ -1

-2x + y < 233

11

22

-1-1

11 33

22

-2-2-3-3 -1-1

TEST: (0,0)

Graph

0 + 0 ≥ -10 ≥ -1

(0,0)

y = - x - 1


Consider the system

x + y ≥ -1

-2x + y < 233

11

22

-1-1

11 33

22

-2-2-3-3 -1-1

Graph

TEST: (0,0)

-2(0) + 0 < 20 < 2

y = 2x + 2

(0,0)

33

11

22

-1-1

11 33

22

-2-2-3-3 -1-1

33

11

22

-1-1

11 33

22

-2-2-3-3 -1-1

x + y ≥ -1 -2x + y < 2


Consider the system

x + y ≥ -1

-2x + y < 2 33

11

22

-1-1

11 33

22

-2-2-3-3 -1-1

SOLUTION:• Lies where the two shaded regions intersect each other.


Consider the system

-2x + 3y < -6

5x + 4y < 12

-1-1

33

11

22

-2-2

Graph

y = x - 223

3322 44-1-1-2-2 11TEST: (0,0)(0,0)

-2(0) + 3(0) < -6

0 < -6 X

X


Consider the system

-2x + 3y < -6

5x + 4y < 12

-1-1

33

11

22

-2-2

Graph

y = - x + 354

3322 44-1-1-2-2 11TEST: (0,0)

5(0) + 4(0) < 12

0 < 12

(0,0)


Consider the system

-2x + 3y < -6

5x + 4y < 12

-1-1

33

11

22

-2-2

Graph

3322 44-1-1-2-2 11

(0,0) SOLUTION:• Lies where the two shaded regions intersect each other.


Consider the system

-2x + 3y < -6

5x + 4y < 12

-1-1

33

11

22

-2-2

Graph

3322 44-1-1-2-2 11

(0,0) NOTE:• All order pairs in dark region are true in both inequalities.


Consider the system

x - 4y ≤ 12

4y + x ≤ 12

(0,0)

101088 1212664422

-2-2

66

22

44

-4-4

-6-6

TEST: (0,0)

(0) - 4(0) ≤ 12 0 - 0 ≤ 12

0 ≤ 12

Graph


Consider the system

x - 4y ≤ 12

4y + x ≤ 12

(0,0)

101088 1212664422

-2-2

66

22

44

-4-4

-6-6

TEST: (0,0)

4(0) + (0) ≤ 12 0 ≤ 12

Graph

HOMEWORK…

Finish pg. 289 #8-16 (solve the system of inequalities by graphing)

#19 and 20.

Problem ModelPatricio’s family, on average, drives their SUV more than twice as many miles as they drive their car. His family’s car emits 0.75 pounds of CO2 per mile and the SUV emits 1.25 pounds of CO2 per mile. Patricio is concern with the environment and convinces his family to limit the total CO2 emissions to less than 600 pounds per month. How many miles can they drive their car and SUV to meet this limit?

xy

= SUV miles= Car miles

> 2yx0.75y 1.25x 600<+

Problem Model

> 2yx0.75y 1.25y 600<+

Problem ModelThe science club can spend at most $400 on a field trip to a dinosaur exhibit. It has enough chaperones to allow at most 100 students to go on the trip. The exhibit costs $3.00 for students 12 and under and $6.00 for students 12 and over. How many students 12 years and under can go if 20 students over 12 go?

xy

= Students 12 and under= Students 12 and over 4y3x 400≤+

yx 100≤+

Problem Model

4y3x 400≤+yx 100≤+

Now you try…

xy

= front-page ads= inside-page ads 1y2x 30≤+

yx 20≤+

The Math Club want to advertise their fundraiser each week in the school paper. They know that a front-page ad is more effective than an ad inside the paper. They have a total of $30 budget for advertising. It costs $2 for each front-page ad and $1 for each inside-page ad. If the club wants to advertise at least 20 times, what are the different possibilities for the number of front-page and inside-page ads.

Now you try…

1y2x 30≤+yx 20≤+

Documents

Bell Work… Graph the following equations. 1. 2. 6 4 2 2 14365 1 3 5 6 4 2 2 14365 1 3 5