Solving Linear Equations and Inequalities in Two Variables
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WARM – UP
Solve each equation for y then graph.
1.
2x + y = 5
2.
2y – x = 6
3.
2x + 3y = 15
GSE Algebra I – 2.6
UNIT QUESTION: How do I justify and
solve the solution to a system of
equations or inequalities?
Standard:
MCC9-12.A.REI.1, 3, 5, 6, and 12
Today’s Question:
How do I know where to shade when
graphing linear inequalities?
Standard:
MCC9-12.A.REI.12
SOLVING LINEAR
SOLVING LINEAR
INEQUALITIES IN TWO
INEQUALITIES IN TWO
VARIABLES
VARIABLES
Coordinate Plane
Half plane
Half plane
A region containing all points
that has one boundary, which is
a straight line that continues in
both directions infinitely.
Steps for Graphing Inequalities
Steps for Graphing Inequalities
1.
Get in slope-intercept form
Get in slope-intercept form
2.
Determine solid ( ≤ ≥ )or dashed line
Determine solid ( ≤ ≥ )or dashed line
(< >)
(< >)
3.
Determine whether to shade above or
Determine whether to shade above or
shade below the line (Test Points)
shade below the line (Test Points)
4.
If the test point is true, shade the half
If the test point is true, shade the half
plane containing it.
plane containing it.
5.
If the test point is false, shade the half
If the test point is false, shade the half
plane that does NOT contain the point.
plane that does NOT contain the point.
Dashed Line
Symbols
Symbols
<
>
Shade Left or Below
Shade right or Above
Solid Line
Symbols
Symbols
Ex. 1
Ex. 2
Ex.3
Ex. 4
SOLVING SYSTEMS OF
SOLVING SYSTEMS OF
LINEAR INEQUALITIES IN
LINEAR INEQUALITIES IN
TWO VARIABLES
TWO VARIABLES
Coordinate Plane
System of Inequalities
System of Inequalities
Graph the
line
and
appropriate
shading
for
each inequality on the
same coordinate plane.
System of Inequalities
System of Inequalities
Remember:
DASHED for < and >,
and
SOLID for
and ≥.
Solution to a System of Linear
Solution to a System of Linear
Inequalities
Inequalities
The solution is the section
where all of the shadings
overlap
.
Sometimes it helps to use different color pencils
for each line and shaded region. It makes it easier
to determine the overlapped shaded regions.
Ex. 5
Ex. 6
Ex. 7
It takes 2 hours to make the blade of a figure
skate. It takes 3 hours to make the blade of a
hockey skate. There is a maximum of 40 hours
per week in which the blades can be made for
both types of skates.
It takes 3 hours to make the boot of a figure
skate. It takes 1 hour to make the boot for a
hockey skate. There is a maximum of 20 hours
per week in which boots can be made for both
types of skates.
Ex. 7
1.
What is the inequality that represents the
time it takes to make the blades of the
two different types of skates?
2.
What is the inequality that represents the
amount of time it takes to make the boots
of the two different skates?
Ex. 7
Graph and consider some constraints to the
problem.
What is the solution of all possible
combinations of figure skates and hockey
skates that can be produced given the
constraints of this situation?
#7’s Graph
Ex. 7’s Solution with Constraints
Ex. 8
Which system of inequalities represents the graph
below?
Ex. 9
Brenda’s high school theater
can seat at most 400 people.
Adult tickets are $5 and
student tickets are $2. The
school must make at least
$1000 for the show to go on.
Which region represents the
possible numbers of adult and
student tickets sold that meet
the given conditions?
Worksheet
Worksheet
Homework
Learn how to solve linear equations and inequalities in two variables by graphing them on a coordinate plane. Understand the steps for graphing inequalities, including determining solid or dashed lines and shading above or below the line. Explore examples and tips for shading regions correctly to find solutions to systems of linear inequalities.
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Presentation Transcript
WARM UP Solve each equation for y then graph. 2x + y = 5 2y x = 6 2x + 3y = 15 1. 2. 3.
GSE Algebra I 2.6 UNIT QUESTION: How do I justify and solve the solution to a system of equations or inequalities? Standard: MCC9-12.A.REI.1, 3, 5, 6, and 12 Today s Question: How do I know where to shade when graphing linear inequalities? Standard: MCC9-12.A.REI.12
SOLVING LINEAR INEQUALITIES IN TWO VARIABLES Coordinate Plane
Half plane A region containing all points that has one boundary, which is a straight line that continues in both directions infinitely.
Steps for Graphing Inequalities 1. Get in slope-intercept form 2. Determine solid ( )or dashed line (< >) 3. Determine whether to shade above or shade below the line (Test Points) 4. If the test point is true, shade the half plane containing it. 5. If the test point is false, shade the half plane that does NOT contain the point.
Symbols Dashed Line < > Shade Left or Below Shade right or Above
Symbols Solid Line
Ex. 1 y 3 x 7 +
Ex. 2 1 3 y x 4
Ex.3 x 2 y 6
Ex. 4 2 x y 4
SOLVING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES Coordinate Plane
System of Inequalities Graph the line and appropriate shading for each inequality on the same coordinate plane.
System of Inequalities Remember: DASHED for < and >, and SOLID for and .
Solution to a System of Linear Inequalities The solution is the section where all of the shadings overlap. Sometimes it helps to use different color pencils for each line and shaded region. It makes it easier to determine the overlapped shaded regions.
Ex. 5 y y 2x 1 2x + 5 +
Ex. 6 y y 2x 1 2x + 5 +
Ex. 7 It takes 2 hours to make the blade of a figure skate. It takes 3 hours to make the blade of a hockey skate. There is a maximum of 40 hours per week in which the blades can be made for both types of skates. It takes 3 hours to make the boot of a figure skate. It takes 1 hour to make the boot for a hockey skate. There is a maximum of 20 hours per week in which boots can be made for both types of skates.
Ex. 7 1. What is the inequality that represents the time it takes to make the blades of the two different types of skates? 2 + x 3 y 40 2. What is the inequality that represents the amount of time it takes to make the boots of the two different skates? 3 x y 20 +
Ex. 7 Graph and consider some constraints to the problem. What is the solution of all possible combinations of figure skates and hockey skates that can be produced given the constraints of this situation?
Ex. 8 Which system of inequalities represents the graph below?
Ex. 9 Brenda s high school theater can seat at most 400 people. Adult tickets are $5 and student tickets are $2. The school must make at least $1000 for the show to go on. Which region represents the possible numbers of adult and student tickets sold that meet the given conditions?
Homework Worksheet
