Lesson Summary



An Inquiry Based Approach to Solving a System of Linear Inequalities

Lesson Summary

1. This four day lesson will guide students through the necessary steps to

2. successfully graph systems of linear inequalities.

Key words used

Solving systems of inequalities.

Background knowledge

1. Number sense and meanings of inequality symbols.

OACS

1. Patterns, Functions and Algebra Standard

1. - Benchmarks D, H – Grades 5-7

2. - Benchmarks F, H – Grades 8-10

3. - Indicators Grade 5 - #4, Grade 6 -#4, Grade 7 - #6,

4. 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.

Plotting inequalities on a number line

Name________________________________________

Goal: This lesson is designed to help you work with and understand basic inequalities.

1. Given the statement: The temperature will be higher than 70 degrees today.

A) Give three possible temperatures that satisfy the statement.

B) We could represent this by using an inequality. (x > 70)

C) 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)

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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.

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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?

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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.

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? |

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|< | | |

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|> | | |

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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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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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)

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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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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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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

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5. Name three ordered pairs on your line.

6. The graph of x = -5 will be a horizontal/vertical (choose one) line.

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 X-intercept 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.

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The graph in the above picture crosses the X-axis at point (5,0) therefore we say the X-intercept 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.

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What conclusions can you draw about (0,3) and (-2,0)?

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?_______

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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 Y-intercept 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 Y-axis. Now connect the dots (remember to extend your line through these points).

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Your Turn!

Graph 10x + 5y = 20

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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?

Activity worksheet.

Graph the following equalities using the X and Y-intercepts.

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3x + 3y = 9

What is your X-intercept?_______

What is your Y-Intercept?_______

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7x – 2y = 14

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What is your X-intercept?_______

What is your Y-Intercept?_______

-4x – 8y = 24

What is your X-intercept?_______

What is your Y-Intercept?_______

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!

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3x + 2y > 6

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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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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.

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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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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).

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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?

Now it's your turn all by yourself.

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6x – 3y > 24

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4x + 2y ≥ -8

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5x -15y < 30

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.

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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?

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.

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7x + 5y < 35

8x – 7y ≤ 56

2x + 3y > -12

5x – 3y ≥ -15

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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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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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