Systems of Linear Inequalities
Date:
Math 11 Notes
Background Information: Graphing Linear Equations
Recall
Most linear equations are written in one of two different forms:
1) y  mx  b
________________________________________ Form
2) Ax  By  C  0
m = ________________, b = ___________________
________________________________________ Form
A = ________________, B, C = __________________
The method used to graph a linear equation depends on the
form given:
Method 1: y  mx  b
-Plot the point (0, b)
-The slope, m, tells us the _________/__________ Use the slope
to create more points from (0, b)
-Draw a straight line through the points. Extend it in both
directions
Example
Graph the linear equations below:
a) y  2 x  3 .
Practice
b) y 
2
x 1
3
Graph the linear equations below:
a) y  x  2 .
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b) y   x  3
2
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Systems of Linear Inequalities
Math 11 Notes
Example
Method 2: Ax  By  C  0
Date:
-Determine the x-intercept by replacing y with 0, and then
solve for x.
-Determine the y-intercept by replacing x with 0, and then
solve for y.
-Plot the x and y intercepts on the grid.
-Draw a straight line connecting the intercepts. Extend it
in both directions.
Practice
Graph the linear equations below:
a) 2 x  y  3  0
b) 3 x  2 y  1  0
Practice
Graph the linear equations below:
a) x  3 y  2  0
b) 4 x  2 y  3  0
Match each linear relation to its graph:
1) 4 x  y  2  0 Graph ______ 2) x  4 y  8  0 Graph ______
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3) y   x  2 Graph ______ 4) y  4 x  2
Graph ______
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Systems of Linear Inequalities
Math 11 Notes
Date:
Linear Inequalities
Linear
Inequality
A linear inequality is similar to a linear equation, except that the
equals sign is replaced with an ________________________________ sign
(>, <, ≥ or ≤).
Recall
> = _________________________
≥ = _________________________
Practice
Solve the following inequalities. Explain the solution.
a) 2x  5  9
b) 5  3x  8
Note
< = _________________________
≤ = _________________________
When you divide an inequality by a negative number, the
direction of the inequality is switched (ie. a ‘less than’ sign
becomes a ‘greater than’ sign and vice versa). When you add or
subtract a negative number, the sign remains the same.
Think
Is the point (2, 1) a solution for the inequality 3 x  2 y  8 ?
Example
You have $100 to buy a mixture of dates and almonds. Dates
cost 32 AED/Kg and almonds cost 90 AED/Kg. Write an
equation that represents the situation where you want to spend
exactly $100.
Practice
Write an inequality that represents the same situation except
that you want to spend less than $100.
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Systems of Linear Inequalities
Date:
Math 11 Notes
Practice
Write an inequality that represents the same situation except
that you want to spend up to, but no more than, $100.
Think
What combination of dates and nuts will satisfy the inequality
above? Give three examples of possible solutions.
Think
Graph the linear equation (from the example part 1). Shade the
side of the line that your three possible solutions fall on (for the
think question above). What do you think this shaded region
represents?
Solution
Region
A solution set is the set of _______ possible solutions. The solution
set (or region) for a linear equation is a_____________. The solution
region for a linear inequality is the area _____________ or ___________
the line (and ____________________________ the line when necessary).
The line of the equation is the ______________________of the solution
region. The boundary line is solid or broken (dotted) according
to the following conditions:
A solid boundary line is used when the equality is included (ie.
the inequality sign is ≤ or ≤).
A dotted boundary line is used when the equality is not included
(ie, the inequality sign is < or >).
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Systems of Linear Inequalities
Date:
Math 11 Notes
Graphing Linear Inequalities
Step 1: Graph the corresponding linear equation using the
method of your choice (ie. one of the methods discussed on pg.
1 and 2, or using a table of values).
Half Plane
The linear equation divides the coordinate plane into two
regions. These are called half planes. The solution region will be
on one side of the line.
Coordinate Plane
Half Plane
Linear Equation
Half Plane
Solution Region
Step 2: To determine which side of the line the solution region is
on, choose a point not on the line and determine if the point
satisfies the inequality (ie. makes the inequality __________). If the
inequality is satisfied, then the region containing the test point
is the _______________________________ region. If it is not satisfied, the
other region is the solution region.
Step 3: Shade the appropriate (solution) region.
Example
Graph and shade the solution region for 2 x  y  6 .
-The line is dotted / solid.
-The test point (0, 0) makes the
inequality true / false.
-The solution region is above / below
the line.
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Systems of Linear Inequalities
Date:
Math 11 Notes
Practice
Graph and shade the solution region for y  3 x  2 .
Step 1: Graph the equation y  3x  2 . Is the line solid or dotted?
Step 2: Choose a test point and plug it into the equation. Use the
results to help you decide which side of the line is the solution
region. Hint: The easiest test point to choose is (0, 0) if it is not
on the line.
Note
If the inequality is of the form y > mx + b or y ≥ mx + b, the
shaded region is always ___________________ the line.
If the inequality is of the form y < mx + b or y ≤ mx + b, the
shaded region is always _______________ the line.
Think
Can you explain why this is true?
Practice
Graph and shade the following inequalities:
a) y  x  1
b) 3x  4 y  12
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Systems of Linear Inequalities
Math 11 Notes
Example
Date:
Sketch the graphs of the following inequalities:
a) y  0
Practice
Fill in the missing inequality signs based on the graphs given:
y _____ x
Practice
c) x  0, y  0
b) x  0
y _____ x
y _____ x
y _____ x
Determine the inequality that is represented by each of the
solution regions shown below:
a)
b)
c)
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Systems of Linear Inequalities
Date:
Math 11 Notes
Continuous
Continuous numbers are a _______________________________ set of
numbers. There is always another number between any two
numbers. Continuous variable are unbroken. For example, a
change in ___________ or temperature is continuous. You cannot
suddenly jump from 20° to 30°.
Think
If the solution of a linear inequality is continuous, what type of
number is the solution set? (whole, integer or real numbers)
Note
So far, all of the examples we have dealt with have been
continuous. We show a continuous solution set by shading the
entire region because any point in the region is a solution.
Discrete
When something is discrete, it consists of separate or
_____________________ parts. Discrete numbers are not continuous,
there are gaps between them. Discrete variables represent
things that are counted, like ________________ or number of objects.
Think
If the solution of a linear inequality is discrete, what type of
number is the solution set likely to be?
Note
There is only one difference when finding the solution region
for a linear inequality that is discrete: instead of shading the
solution region we ‘stipple’ the solution points (use a dot to
indicate all of the possible solutions).
Example: x  y  3  0
x, y ε W (whole #s)
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Systems of Linear Inequalities
Date:
Math 11 Notes
Example
Graph the solution region for the inequality:
-3y + 6 ≥ -6 + y; x and y ε I (integers)
Practice
Graph the solution region for the inequality:
x – 2 > y; x and y ε W (whole numbers)
Practice
Graph the solution region for the inequalities:
x > 0; y < 0, x and y ε W
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Systems of Linear Inequalities
Date:
Math 11 Notes
Modelling Problems Using Linear Inequalities
Example
A sports store makes $100 on every pair of skis it sells and $120
on every snowboard it sells. The manager’s goal is to make
more than $600 dollars per day selling skis and snowboards.
a) Write a linear inequality that represents this situation.
b) Give an example of a combination of ski and snowboard sales
will meet or exceed the manager’s daily goal of $600 profit.
Constraint
Constraints describe limiting conditions or ________________________
on the situation given. For example, if ‘x’ represents time, then
x ______ 0 because time cannot be ______________________________.
c) What are the constraints in the situation above?
d) What set of numbers do the solution points belong to?
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Systems of Linear Inequalities
Date:
Math 11 Notes
d) Graph and shade the solution region for the situation.
Practice
Sarah likes going to the movies and going to Saadiyat Beach. It
costs 35 AED to see a movie and 25 AED for entrance to the
beach. She budgets 175 AED per month for these activities.
a) Write a linear inequality that describes the situation. Make
sure you define (explain the meaning of) each variable.
b) What are the constraints on the variables?
c) Graph and shade the solution region for the inequality.
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Systems of Linear Inequalities
Date:
Math 11 Notes
Solving Systems of Linear Inequalities
Recall
The solution of a system of linear equations is a point that
makes both linear equations ___________. On a graph, the solution
is the point where the two lines ___________________________.
Example
Find the solution of the system of equations both algebraically
and graphically:
y = 2x - 1
y = ½x + 2
Systems
of Linear
Inequalities
Example
A system of linear inequalities is a set of ___________________________
linear inequalities. When graphed on the same coordinate
plane, the _________________________________ of their solution regions
is the solution of the system.
The light grey regions show the solution
regions of y ≤ -x-5 and y ≥ 2x +5. The
darker region is the area where the
solution regions overlap. This is the
solution region for the system of
inequalities (ie. the region of points
that are true for both inequalities).
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Systems of Linear Inequalities
Date:
Math 11 Notes
Practice
What system of inequalities is represented by the graph below?
What set of numbers do the solution points belong to?
Practice
Find the solution region of the system below by graphing:
y > 2x – 3
y ≤ -x +5
Practice
Find the solution region of the system below by graphing:
4x + 2y - 8 ≥ 0
3x – 4y – 4 < 0
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Systems of Linear Inequalities
Math 11 Notes
Example
Give the inequalities: 1) x – y > 4
Date:
2) x + y < 6
a) Graph and shade the solution
region for the system.
b) Give two possible solutions to
this system of inequalities.
Think
Is the intersection point (5, 1) included in the solution? Explain.
Practice
A region is defined by the following inequalities:
1) x ≥ 0
2) y ≥ 0
3) x +y ≤ 4
4) 3x + 8y ≤ 24
a) Graph and shade the sol. region. if
x and y are whole numbers.
b) Is x = 1, y = 3 a solution to the
system of inequalities?
c) List all possible solutions to the system (remember that x and
y must be whole numbers).
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Systems of Linear Inequalities
Date:
Math 11 Notes
Modelling Problems Using Systems of Linear Inequalities
Write three linear inequalities that represent the following
statements:
Example
The sum of two whole numbers is less than 10.
Practice
The difference between two girls’ ages is at least 4.
Example
A gardener uses up to 30 square meters to plant peas and
carrots.
Example
A company makes aluminum and fibreglass boats. The factory
can make a maximum of 20 boats per day. There is a higher
demand for fibreglass boats, so at least 5 more fibreglass than
aluminum boats are made.
a) Write a system of inequalities to represent this situation.
b) What are the constraints on the variables?
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Systems of Linear Inequalities
Date:
Math 11 Notes
c) Graph the solution region for the system. Explain what the
solution region tells you based on the situation given.
d) Is the intersection point of the lines a solution? Explain.
Example
Erin is mixing blue and yellow dye to make green. She needs at
least 100 ml of green dye. She wants to use at least 20 ml more
blue dye than yellow. What combinations of dye are possible?
a) Write a system of linear inequalities to represent the
situation.
b) What constraints are on the variables? What set of numbers
do the solutions belong to?
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Systems of Linear Inequalities
Math 11 Notes
Date:
Optimization Problems
Optimization
Problem
An optimization problem is a problem where a quantity must be
_______________________________ or ___________________________ following
a set of guidelines or ______________________ (ie. a limiting condition
on the situation that is represented by a linear inequality).
Objective
Function
The objective function is the equation that represents the
relationship between the two variables in the system of linear
inequalities. The objective function is the quantity we are trying
to ______________________________________ (maximize or minimize).
Optimal
Solution
The optimal solution is the point in the solution set that
represents the _________________________________________________ value
of the objective function.
Feasible
Region
The feasible region is the ____________________________________________
for the system of linear inequalities. The optimal solution is in
the feasible region.
Example
A toy company makes two types of toy cars: race cars and SUVs.
No more than 40 race cars and 60 SUVs can be made per day.
However, the company must make at least 70 cars in total each
day. It costs $8 to make a race car and $12 to make a SUV.
a) Write three linear inequalities to represent the following:
i) Total number of race cars that can be made per day.
ii) The total number of SUVs that can be made per day.
iii) The total number of vehicles that can be made per day.
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Systems of Linear Inequalities
Date:
Math 11 Notes
b) What are the restrictions on the variables?
c) What quantity needs to be optimized in this situation? Would
you want to maximize or minimize it? Write an equation to
represent how the two variables relate to this quantity. Graph
the system of linear inequalities.
d) Choose some solutions in the solution region and check the
value of C for each. Do you notice a pattern?
e) What points in the solution region result in an optimal
solution (either a maximum or a minimum value for C)?
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Systems of Linear Inequalities
Math 11 Notes
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Finding Optimal Solutions
The optimal solutions occur at the intersection points of the
boundary lines (the lines corresponding to the inequalities).
Note
Not all intersection points are optimal solutions, but all optimal
solutions are intersection points.
Vertex
The intersection points of the boundaries are called the vertices
or corners, or the feasible region. (Vertices = plural of vertex)
If one or more of the boundary lines is broken (not included in
the solution region) then the optimal point is the point as close
as possible to the intersection. If the intersection point is not a
whole number, but the solutions must be whole numbers, then
the optimal point is the closest possible whole number point to
the intersection point.
Example
Imagine that we are construction contractors and have been
given a contract to build a fence. The fence will be no longer
than 50 yards. It will be built with a combination of 4 inch
narrow boards and 8 inch wide boards. There must be at least
100 wide boards and no more than 80 narrow boards. The
narrow boards cost $3 each and the wide boards cost $5 each.
Determine the maximum and minimum costs for the lumber to
build the fence.
a) Create the system of inequalities. What are the constraints on
the variables?
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Systems of Linear Inequalities
Date:
Math 11 Notes
b) Create the objective function (the equation that represents
the relationship between the variables and the total cost of the
boards for the fence).
c) Graph the solution region. List the vertices (the intersection
points of the boundary lines).
d) Test the intersection points to identify the optimal solutions
(plug the numbers into the cost equation to see which point
gives the maximum cost and which point gives the minimum
cost).
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