6.6 Solving System of Linear Inequalities:
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Transcript of 6.6 Solving System of Linear Inequalities:
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
12345678910
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.