Project AMP
Dr. Antonio R. Quesada Director, Project AMP
An Inquiry Based Approach to Solving a System of Linear Inequalities
Lesson Summary
This four day lesson will guide students through the necessary steps to
successfully graph systems of linear inequalities.
Key words used
Solving systems of inequalities.
Background knowledge
Number sense and meanings of inequality symbols.
OACS
Patterns, Functions and Algebra Standard
- Benchmarks D, H – Grades 5-7
- Benchmarks F, H – Grades 8-10
- Indicators Grade 5 - #4, Grade 6 -#4, Grade 7 - #6,
Grade 8 - #7, 9,10, Grade 9 - #6, 9, Grade 10 - #7, 11
Learning objectives
This lesson is designed to help the students discover and work with basic
inequalities. The lesson is designed to guide the students through graphing
inequalities on a number line and using X and Y-intercepts to graph linear
equations on the coordinate grid. The lesson will then have the students graph
linear inequalities and correctly shade the solution area on the coordinate grid.
This will then lead into having the student use X and Y-intercepts to graph a
system of linear inequalities on the coordinate grid and correctly shade the
solution area. Finally, the lesson will conclude with the students having to locate
a treasure chest by using a system of linear inequalities.
Materials needed
Worksheets
Colored pencils
Procedures
Place students in groups, 2-4 per group. Have a class discussion after students
have completed the activities.
Assessment
Assessments will be included throughout and after each lesson. In addition, there
will be a treasure hunt activity at the end of the lesson that will act as a final
assessment of the lesson.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Plotting inequalities on a number line
Name________________________________________
Goal: This lesson is designed to help you work with and understand basic inequalities.
1.
A)
B)
C)
Given the statement: The temperature will be higher than 70 degrees today.
Give three possible temperatures that satisfy the statement.
We could represent this by using an inequality. (x > 70)
We could also represent this with a number line graph (Notice the open circle
above 70. This indicates that 70 is not included in the values represented on
the graph since we only wanted temperatures larger than 70)
○
<-----|-----|-----|-----|-----|-----|-----|----->
40
50
60
70
80
90
100
2. Given the statement: To ride the Little Dipper you must be under 48 inches
tall.
A) Give three possible heights that satisfy this statement
B) Write an inequality to represent the statement.
C) Notice that the number line graph is going the opposite direction since the
inequality is different.
○
<-----|-----|-----|-----|-----|-----|-----|----->
44
46
48
50
52
54
56
3. Given the statement: The temperature will be 70 degrees or warmer today.
A) Give three possible temperatures that satisfy the statement.
B) We could represent this by using an inequality. (x > 70)
C) What is different between the graph of this statement and the statement in
#1?(look closely at the circles) Why do you think this difference is necessary?
●
<-----|-----|-----|-----|-----|-----|-----|----->
40
50
60
70
80
90
100
4. Given the statement: To ride the Little Dipper you must be 48 inches tall or
shorter.
A) Give three possible heights that satisfy this statement
B) Write an inequality to represent the statement.
C) If you were drawing the number line graph, would you use an open or closed
circle? Why?
D) Would your line go to the right or left? Why?
E) Draw the number line graph.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
<-----|-----|-----|-----|-----|-----|-----|----->
44
46
48
50
52
54
56
5. If x > 5, what does this mean? What x values make the statement true?
6. Why is 5 not included in your list?
7. If x < 2, what does this mean? What x values make the statement true?
8. Why is 2 included in this list?
9. Complete the following table
Symbol
Meaning
Open or closed Circle?
>
<
>
>
10. To graph x > 4 on a number line, we would need to show all the numbers that
are greater than but not equal to 4. Draw a number line graph for x > 4
<-----|-----|-----|-----|-----|-----|-----|----->
0
1
2
3
4
5
6
11. Graph x < 2 on a number line.
<-----|-----|-----|-----|-----|-----|-----|----->
0
1
2
3
4
5
6
12. Plot the following ordered pairs on the same coordinate grid and then connect
the points with a straight line. (2,6), (-3,6), (-7,6), (3,6)
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
What do you notice about your line? __________________________________
What do you notice about your ordered pairs? ___________________________
13. Graph y = 6 on the coordinate grid.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
14. Compare your graphs from #5 and #6. What do you notice about them?
_________________________________________________________
What type of line do you get when all your y values are the same? ____________
15. Plot the following ordered pairs on the same coordinate grid and then connect
the points with a straight line. (3,7), (3,-4), (3, -5), (3,4)
What do you notice about your line? ___________________________________
What do you notice about your ordered pairs? ____________________________
16. Graph x = 3 on the coordinate grid.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
17. Compare your graphs from #7 and #8. What do you notice about them?
_________________________________________________________
What type of line do you get when all your x values are the same? ____________
Look at the following graphs and answer the questions.
A) List 3 ordered pairs for each graph
B) Write an equation of the line represented by the graph
18.
19.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Day 1 Recap Worksheet
1. Write an inequality for the statement The temperature will be cooler than 50
degrees tonight.
2. Draw a number line graph to represent the statement in question 1.
<-----|-----|-----|-----|-----|-----|-----|----->
3. Draw a number line graph to represent the inequality x > -4
<-----|-----|-----|-----|-----|-----|-----|----->
4. On a coordinate grid, graph the line y = -2
5. Name three ordered pairs on your line.
6. The graph of x = -5 will be a horizontal/vertical (choose one) line.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Graphing the standard form of linear equations on the coordinate grid.
Name_______________________________________
Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph
linear equalities on the coordinate grid.
Intercepts are where one or more objects cross each other. When we talk about the Xintercept we are concerned with the point where the graph crosses the X-axis. Likewise,
when we talk about the Y-intercepts of a graph, we are concerned with the point where
the graph crosses the Y-axis.
The graph in the above picture crosses the X-axis at point (5,0) therefore we say the Xintercept is 5. It crosses the Y-axis at point (0,-4) and we say that the Y-intercept is -4.
Using your knowledge of plotting points on the coordinate grid, plot (0,3) and (-2,0).
Draw a line through these points.
What conclusions can you draw about (0,3) and (-2,0)?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Where is the graph crossing the X-axis?________ What is the X-intercept?_______
Where is the graph crossing the Y-axis?________ What is the Y-intercept?_______
Using the above picture, what do you notice about the coordinates where the graph
crosses the X and Y-axis?
You should have noticed that at the X-intercept the Y value is always zero and at the Yintercept the X value is always zero. Now let's use this information to quickly construct
graphs of linear equations that are in the written in the standard form.
Take a look at 4x + 6y = 12. In order to find the X-intercept we know that the Y value
must equal zero. Therefore, if we substitute zero in for y we now have: 4x + 6(0) = 12.
This then gives us 4x = 12. Solving for x we get x = 3. That's our X-intercept!
Let's find the Y-intercept.
Remember that when we are looking for an intercept, one value must always equal zero.
Since we are looking for the Y-intercept our X value must equal zero.
Therefore, 4(0) + 6y = 12. This gives us 6y = 12, and y = 2. That's our Y-intercept.
Now let's graph the line. Our X-intercept is 3 so we plot a point at 3 on the positive side
of the X-axis and our Y-intercept is 2 so we plot a point at 2 on the positive side of the Yaxis. Now connect the dots (remember to extend your line through these points).
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Your Turn!
Graph 10x + 5y = 20
What is the X-intercept? (hint: where does
it cross the X-axis?) ______
What is the Y-intercept?______
Did you plot the points correctly and
remember to extend your line through the
points?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Activity worksheet.
Graph the following equalities using the X and Y-intercepts.
3x + 3y = 9
What is your X-intercept?_______
What is your Y-Intercept?_______
7x – 2y = 14
What is your X-intercept?_______
What is your Y-Intercept?_______
-4x – 8y = 24
What is your X-intercept?_______
What is your Y-Intercept?_______
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Graphing the standard form of linear inequalities on the coordinate grid.
Name______________________________________
Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph
linear inequalities on the coordinate grid and correctly shade the solution area.
When we are graphing linear inequalities that are written in the standard form, we follow
the same format of using the X and Y-intercepts. There are several very important
differences with the actual graph though. Let's take a look!
3x + 2y > 6
On the right is the graph of the above linear
inequality. What do you notice about the
graph?
4x – 12y ≥ 24
On the right is the graph of the above linear
inequality. What do you notice about the
graph?
What is different between this graph and the
first graph?
Just like when we graphed numbers on a number line and had closed and open circles
depending on the inequality used, the same rule applies here. Instead of circles though,
now we have lines. When we are using the < and > inequalities, our graph is a dotted
line. When we are using ≤ and ≥ inequalities, our graph is a solid line. The solid line
indicates that we include the points on the line.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Now you try it!
-2x + 4y < 12
What is your X-intercept?________
What is your Y-intercept?________
Did you remember to use the correct line
style?
Let's go take a look at the following inequality.
3x + 2y ≥ 6
Notice that graph cuts the coordinate plane
into two pieces. One above the graph and
one below the graph. Remember that when
we are dealing with inequalities, only certain
values actually "work" in the inequality.
These values that work, make the inequality
true and those values that don't work make
the inequality false. In order to find out
which work and which don't, we pick test
points. Test points are any points that lie
above or below the graph. Let's pick the
point (0,0) and find out if it works or not in the inequality.
If we substitute (0,0) into the inequality we get 3(0) + 2(0) ≥ 6. Solving we get 0 + 0 ≥ 6,
which gives us 0 ≥ 6. Obviously this is false and therefore (0,0) does not work in the
inequality. Notice that (0,0) is located below our graph.
Now let's try a point above our graph. How about (4,4) (it doesn't matter what point just
so long as it is above the graph). If we substitute (4,4) into our inequality we get 3(4) +
2(4) ≥ 6. Solving we get 12 + 8 ≥ 6 and this gives us 20 ≥ 6. Obviously true.
What do you think will happen if you pick any point below the graph?
What do you think will happen if you pick any point above the graph?
What do you think will happen if you pick any point on the graph?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
The correct representation of the previous inequality is graphed below.
Wow, now some of the coordinate plane is
shaded.
Why do you think just the area above the graph
is shaded?
If you said because that is where all the true
value are, you're correct. When we are
graphing inequalities, you must shade the area
of the coordinate plane that contains the ordered
pairs that work in the inequality.
Let's try one more together. How about -2x + 6y ≤ 18.
Based on the inequality, what type of line will we have (solid or dotted)?___________
Start with the X and Y-intercepts.
X-intercept is _______________________
Y-intercept is _______________________
Now let's plot those points and connect the dots with a _____________ line (solid or
dotted).
Pick a test point. I always like to use (0,0) when
it is available.
If we substitute (0,0) into our inequality, do we
end up with a true or false answer?
What side of the coordinate plane should we
shade, above or below?
Did you shade it?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Now it's your turn all by yourself.
6x – 3y > 24
4x + 2y ≥ -8
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Dr. Antonio R. Quesada Director, Project AMP
5x -15y < 30
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Graphing and solving a system of linear inequalities on the coordinate grid.
Name__________________________________________
Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph
a system of linear inequalities on the coordinate grid and correctly shade the solution
area.
In the previous lesson you learned how to graph and shade a linear inequality. Now you
will learn how to graph and shade a system of linear inequalities. A system of linear
inequalities is more than one linear inequality that will have common solutions. Through
graphing and shading you will be able to discover the area that represents the common
solutions. Let's get started!
As mentioned before, a system is more than one inequality. Let's start with two linear
inequalities.
3x + 9y > 27
4x -2y ≤ 12
Approach these inequalities one at a time.
Find the X and Y-intercepts of the first
problem.
X-intercept_______
Y-intercept_______
Plot those points and draw your graph. Did
you remember to use the correct type of line?
Now pick a test point and shade the correct
portion of the coordinate plane with the blue colored pencil.
Time to graph the next inequality. Start with the intercepts.
X-intercept_______________
Y-intercept_______________
Plot the points, draw your graph with the correct type of line and pick a test point that is
either above or below the linear inequality you just graphed, not the one you graphed
earlier. Shade the correct side with the yellow colored pencil.
Are there any areas that contain both blue and yellow?
- 18 -
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
In order to solve a system, the solution must satisfy each equation in the system. In our
case, the inequalities must both be true. Let's pick a test point (how about (0,0)) and
substitute it into both inequalities. The first inequality will give us 0 > 27 and the second
will give us 0 ≤ 12. The first is false and the second is true.
What do you notice about the shading at (0,0)?
Now let's try (1,8). The first inequality gives us 75 > 27 and the second inequality gives
us -12 ≤ 12. Both of these are true!
What do you notice about the shading at (1,8)?
What conclusions can you draw about shading and solutions to the system?
Do you think anything changes if there are three, four, five or more inequalities in our
system? If you answered no, you're right. Besides a little more work, nothing changes.
Let's try a system with 4 inequalities.
7x + 5y < 35
8x – 7y ≤ 56
2x + 3y > -12
5x – 3y ≥ -15
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
An archeologist has discovered a series of coordinates that may
represent the location of buried treasure. After deciphering the
coded messages, it seems that the only true coordinates of the
treasures are those coordinates that lie within the solution of the
systems of inequalities.
Potential Treasures may be located at the coordinates designated on the map
below. Record the possible locations of the treasures on the lines provided.
1. ______________
2. ______________
3. ________________ 4._____________
5. ______________
6. ______________ 7. ________________ 8. _____________
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
The solution to the following system of inequalities is the location of the
treasure. You will need to graph the inequalities on the provided grid. The
treasure will be located at the point that lies within the solution of the system
of inequalities.
2x + 3y < 18
-4x – 4y < 8
-3x + 4y < 12
4x – 5y < 20
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