AM30S | Systems of linear inequalities
Lesson #1
p. 1 of 1
L1 Linear systems review
REVIEW
Slope-intercept form of equation:
y = mx + b
where m is the ______________________ and b is the __________________________ of the line.
Slope =
€
rise
vertical change
y −y
=
= m= 2 1
run horizontal change
x 2 − x1
GRAPHING
Graph the following equations on the grid below.
6
(a) y = 2x + 3
y
4
2
(b) y = –½x – 2
-6
(c) 0 = 3x + y – 2
6
-4
-2
2
-2
-4
-6
INTERCEPTS
Find the x-intercept and y-intercept for (a) above.
4
AM30S | Systems of linear inequalities
Lesson #2
p. 1 of 3
L2 Linear inequalities intro
BASIC INEQUALITIES
Q:
How old must you be to drive in Manitoba?
A:
At least 16… so as a formula we could say: A ≥ 16
More examples:
I want to find a job that pays more than minimum wage.
I have to spend less than $50 this week.
I must make $600 or more per month in order to afford my car.
I need to get at least 40% on my exam to pass the course.
→ What is a linear inequality?
A linear inequality is a relationship between two linear expressions in which one expression is less
than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) the other
expression.
INEQUALITY RULES
1. Use a solid line if it is ≥ or ≤ and a dotted line if it is > or <.
2. Shade above the line if it is > or ≥ and shade below the line if it is < or ≤
(later you will learn how to test a point for this to check)
Examples: Specify if a solid or dotted line AND whether it is shaded above or below the line.
y < –2x – 1
y ≤ 8 – x + 2y
y ≥ 15 + 3x
y>5–x
AM30S | Systems of linear inequalities
Lesson #2
p. 2 of 3
3. When multiplying or dividing by a negative value, you must switch the direction of the
inequality symbol!
Ex:
–3y > 6x – 3
–3y > 6x – 3
–3 –3 –3
y < –2 + 1
Ex: y ≥ x + 2
Which points are on either side?
x
y
0
4
1
2
-3
1
CHOOSE A “TEST POINT”:
Which sets of points, half plane (region on one side of the line), makes the inequality true?
y ≥ x+2
test (0, 0)
0 ≥ 0+2
0 ≥ 2
Is 0 ≥ 2? No! Then the point (0, 0) is not in the zone that satisfies this inequality. Cross of this side
because it’s no good! And shade reverse side.
USING THE TI-83
Find the solution to the following –2x + 5y ≥ 10
- sometimes written { (x,y) | –2x + 5y ≥ 10, x∈R, y∈R }
1. Get into “y = “ form:
5y ≥ 2x + 10
5
5
5
y ≥ 2x + 2
5
2. Enter into “y=” graph
3. Use test point (0, 0) to determine the zone that satisfies the inequality
4. Press “y=” key and cursor over to the left and select the shade for above or below (enter)
** repeatedly clicking the shade button will darken that particular zone
AM30S | Systems of linear inequalities
Lesson #2
p. 3 of 3
WORD PROBLEMS
Donna wants to buy nuts and raisons. Raisins are $3 per pound and nuts are $4 per pound. She
wants to spend less than $50.
(a) Write a formula to represent the relationship.
r = raisins and n = nuts
3r + 4n ≤ 50
(b) What are the restrictions on n and r?
{ (n, r) | 3r + 4n ≤ 50, r∈R, y∈R }
(c) What are two combinations of raisins and nuts that make sense? Explain your choice.
(graph… do test points)
(d) Which combinations would cost exactly $50?
This means on the line!
DAILY ASSIGNMENT
Page 303 #1 – 3, 4, 6ae, 7, 8, 12
AM30S | Systems of linear inequalities
Lesson #3
L3 Graphing systems of inequalities
INEQUALITY RULES
1. Solve in terms of y (“y=…”)
2. Graph the line as if it were an equal sign and not >, <, ≥, or ≤.
3. Use a solid line if it is ≥ or ≤ and a dotted line if it is > or <.
4. Shade above the line if it is > or ≥ and shade below the line if it is < or ≤
GRAPH LINEAR INEQUALITIES
1. Graph 2x + y < 4
2. Graph –6x + 2y ≥ –8
p. 1 of 3
AM30S | Systems of linear inequalities
Lesson #3
EXAMPLE: Graph the following system of inequalities
The area that is shaded by BOTH the graphs is the solution.
x + y > –1
and
Ex) 2x – y ≤ –3
3x – 2y > 4
and
x ≤ –2
Ex) Given the following diagram, write the system of inequalities that represent it.
p. 2 of 3
AM30S | Systems of linear inequalities
Lesson #3
p. 3 of 3
AND/OR
Some questions ask us to graph linear inequality A “and” linear inequality B
This would indicate that they are looking for only the intersecting points – or the shaded areas
common to both.
Similarly, if you wanted a car wit ha sun rood AND heated seats, that limits your options.
Ex) Graph for x + y > 6 and 2x – 2y < 5
Ex) Graph the inequalities: x – 2y < 5 or 2x + y > 3
AM30S | Systems of linear inequalities
Lesson #4
p. 1 of 2
L4 Optimal problems: Creating the model
Optimization problems: Problems where quantity must be maximized or minimized following a set
of guidelines or conditions.
Constraints: Limits on variable.
Feasible region: The solution region for a system.
Objective function: Max/min equation (what you want: max or min)
1. Look for quantities that must be optimized (words: maximize, minimum, largest, smallest,
greatest, least…).
2. What are your variable? Name restrictions of these variables.
3. Write a system of linear inequalities (constraints).
4. Write an objective function.
5. Graph.
EXAMPLE 1
Three teams are traveling to a volleyball tournament in cars and minivans.



Each team has no more than 2 coaches and 14 athletes.
Each car can take 4 members and each minivan can take 6 members.
No more than 4 minivans and 12 cars are available.
The school wants to know the combination of cars and minivans that will require the minimum and
maximum number of vehicles. Create a model to represent the situation.
AM30S | Systems of linear inequalities
Lesson #4
p. 2 of 2
EXAMPLE 2
A refinery produces oil and gas.

At least 2 L of gasoline is produced for each litre of heating oil.

The refinery can produce up to 9 million litres of heating oil and 6 million litres of gasoline
each day.

Gasoline is projected to sell for $1.10 per litre. Heating oil is projected to sell for $1.75 per
litre.
The company needs to determine the daily combination of gas and heating oil that must be
produced to maximize revenue. Create a model to represent this situation.
TO DO: p. 330 #1 – 3, 5, 6
AM30S | Systems of linear inequalities
Lesson #5
p. 1 of 1
L5 Optimal problems: Exploring solutions
Optimal solution: Solution set that represents the maximum or minimum value of the objective
function.
Pg 332 Example
Do the reflecting questions together on p. 333 to explore solutions.
p. 334 # 1 – 3