6.6 Solving System of Linear Inequalities:

System of Linear Inequalities: Two or more linear inequality equations.

Solution of a System of Linear Equations:Any ordered pair that makes all the equations in a system true.

Always Remember:

GOAL:

Y> x + 1

Y< x + 1

> : Dashed line and shade above

< : Dashed line and shade below

Y≤ x + 1

Y ≥ x + 1

≤ : Solid line and Shade below

≥ : Solid line and shade above

Y≤ x + 1

Y> x + 1

Y< x + 1

Y ≥ x + 1

SOLVING A SYSTEM BY GRAPHING: To solve a system of inequalities we must:

1) Write the equations in slope-intercept form (y □ mx+b)

2) Graph the equations and shade

3) Find an ordered pair point inside the shaded intersection region 4) Check

Ex: What is the solution of the system? Use a graph to check your answer.

2 42

x yy x

http://www.meta-calculator.com/online/

SOLUTION:

22 y xy x

1) Write the equations in slope-intercept form (y□mx+b)

22 44 y xx y

SOLUTION: 2y x

2) Graph the equations 2 4y x

Dashed line and shade down Solid line and shade up

SOLUTION: 2y x

2) In the same graph: 2 4y x

SOLUTION: 2y x

3) Find the solution 2 4y x

Looking at the graph, we see that any point in the double shaded region will be a solution, say:

(0, 3)

SOLUTION:

2y x

4) Check

2 4y x

We know that (0,3) is the solution from our graph.

3 2( 40)

3 0 4 3 4 TRUE

3 20 3 2 TRUE

YOU TRY IT: What is the solution of the system? Use a graph to check your answer.

22 3

x yy x

SOLUTION:

3 32 2y x y x

1) Write the equations in slope-intercept form (y=mx+b)

22x yy x

SOLUTION: 3 2y x

2) Graph the equations 2y x

Dashed line and shade down Dashed line and shade down

SOLUTION: 3 2y x

2) In the same graph: 2y x

SOLUTION: 3 2y x

3) Find the solution 2y x

(2,-1)

Looking at the graph, we see that any point in the double shaded region will be a solution, say:

SOLUTION:

3 2y x

4) Check

2y x

We know that (2,-1) is the solution from our graph.

21 2

1 4 TRUE

3( )1 22 1 6 2

1 4 TRUE

Real-World: You are planning what to do after

school. You can spend at most 6 hrs. daily playing the trumpet and doing homework. You want to spend less than 2 hrs. playing the trumpet. You must spend at least 1.5 hrs. on homework. What is a graph showing how much you can spend your time?

Real-World(SOLUTION):You are planning what to do after school. You

can spend at most 6 hrs. daily playing the trumpet and doing homework. You want to spend less than 2 hrs. playing the trumpet. You must spend at least 1.5 hrs. on homework. What is a graph showing how much you can spend your time?

At most 6 hrs x + y ≤ 6

Less than 2 hrs trumpet x < 2 hrs

At least 1.5 hrs homework y ≥1.5 hrs

SOLUTION: x + y ≤ 6 y ≤ - x + 6

1 2 3 4

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5 6

SOLUTION: x < 2

1 2 3 4

12345678910

5 6

SOLUTION: Y ≥1.5

1 2 3 4

12345678910

5 6

SOLUTION:

y ≤ - x + 6

1 2 3 4

12345678910

5 6

x < 2

Y ≥1.5

All together in the same graph:

SOLUTION:

y ≤ - x + 6

1 2 3 4

12345678910

5 6

x < 2

Y ≥1.5

Any point in the red shaded area are solutions to the problem

YOU TRY IT: You have a job mowing the lawn for

$10 per hour. You also have another job playing for parties and you charge $12 per hour. You need to earn at least $350 to buy a new instrument but you cannot work more than 35 hrs per week. You must work a minimum of 10 hrs playing at parties. What is a graph showing how many hours per week you can work at each job?

YOU TRY IT: (SOLUTION)

mowing the lawn

earn at least $350

X

playing at parties y

no more than 35 hrs per week

10x + 12y ≥ 350

minimum of 10 hrs playing

x + y ≤ 35

y ≥ 10

Given info:

SOLUTION: y ≥- x + 35

10 20 30 40

10

20

30

40

50

60

50 6

10x + 12y ≥ 350

SOLUTION:

10 20 30 40

10

20

30

40

50

60

50 6

x + y ≤ 35 y ≤ - x + 35

SOLUTION:

10 20 30 40

10

20

30

40

50

60

50 6

y ≥ 10

SOLUTION: All in one graph

10 20 30 40

10

20

30

40

50

60

50 6

y ≥ 10

y ≤ - x + 35

y ≥- x + 35

SOLUTION: All in one graph

10 20 30 40

10

20

30

40

50

60

50 6

y ≤ - x + 35 y ≥- x + 35

y ≥ 10

Any point inside the red region is a solution

VIDEOS: Graphing Inequalities

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-linear-inequalities/v/solving-and-graphing-linear-inequalities-in-two-variables-1

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-linear-inequalities/v/graphing-inequalities

CLASSWORK:

Page 398-401

Problems: As many as needed to master the

concept.