Graphing Solutions of Equations and Inequalities
Graphing Solutions of Equations and Inequalities
Objectives
In this unit, students will solve equations and inequalities and learn how to represent these solutions as lines as well as shaded regions of the Cartesian plane. Students will:
- graph lines and learn that all the points on a line represent possible solutions to the linear equation.
- learn how to graph systems of inequalities and determine which region on a graph represents the solution to the system.
Essential Questions
- How do you write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities?
- How do you write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- http://www.math.com/school/subject2/lessons/S2U4L3GL.html
- https://www.purplemath.com/modules/ineqgrph.htm
- https://www.youtube.com/watch?v=lxCoahmymqw
- http://webmath.com/plotineq.html
- http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut14_lineargr.htm
- http://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php
- http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/system-linear-inequality.php
- http://www.freemathhelp.com/systems-inequalities.html
- https://www.purplemath.com/modules/syslneq.htm
Formative Assessment
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View
Multiple-Choice Items:
Use the following situation for questions 1, 2, and 3.
Henry is a carpenter. It takes Henry 3 hours to build a cabinet, and 4 hours to build a drawer. He works for 60 hours each week.
1. What is the constraint in this problem?
A
It takes Henry 3 hours to build a cabinet.
B
It takes Henry 4 hours to build a drawer.
C
He works for 60 hours each week.
D
There is no constraint in this problem.
2. Which equation represents this situation, if x represents the number of cabinets Henry makes and y represents the number of drawers Henry makes?
A
3x + 4y = 60
B
4x + 3y = 60
C
60x + 4y = 3
D
3x + 60y = 4
3. The line segment that represents all the possible solutions to this situation has which of the following endpoints?
A
(3, 0) and (0, 4)
B
(0, 3) and (4, 0)
C
(15, 0) and (0, 20)
D
(0, 15) and (20, 0)
4. Harris has $60. He can spend it at an Internet café, which costs $1/minute, or at a video game arcade, which costs $2/minute. Which inequality represents this situation, where x is the number of minutes Harris spends at the Internet café and y is the number of minutes Harris spends at the video game arcade?
A
x + 2y ≥ 60
B
x + 2y ≤ 60
C
2x + y ≥ 60
D
2x + y ≤ 60
5. What best describes the line that would be graphed to solve the equation 4x + 3y ≥ 12?
A
A solid line with a slope of
B
A solid line with a slope of
C
A dashed line with a slope of
D
A dashed line with a slope of
6. Which of the following is not a valid test point to determine which side of the line to shade on for the equation x + 8y < 25?
A
(0, 0)
B
(1, 3)
C
(1, 8)
D
(3, 1)
Use the following system of inequalities for questions 7, 8, and 9.
7. Which of the following is a solution to the above system?
A
(0, 0)
B
(1, 2)
C
(3, 3)
D
(4, 1)
8. In which quadrant is the solution to the above system?
A
I
B
II
C
III
D
IV
9. If the inequality x ≤ 5 in the above system was replaced by x ≥ 5, how would the region representing the solution to the system change?
A
It would not change the solution at all.
B
There would not be a solution to the system.
C
The solution would change, but still be a quadrilateral.
D
The solution would change and would be a triangle.
Multiple-Choice Answer Key:
1. C
2. A
3. D
4. B
5. B
6. B
7. D
8. A
9. D
Short-Answer Items:
10. At a grocery store, peanuts are $2 per pound and walnuts are $5 per pound. Joseph has $30 to spend. Write an equation that represents this situation. Clearly identify the two variables, what each variable represents, and the constraint.
11. Graph the solution to the inequality 3x + 4y ≤ 24.
12. Given a system of four inequalities, explain the steps you would use to find the solution to the system. Your steps should include every detail necessary to arrive at a correct solution.
Short-Answer Key and Scoring Rubrics:
10. At a grocery store, peanuts are $2 per pound and walnuts are $5 per pound. Joseph has $30 to spend. Write an equation that represents this situation. Clearly identify the two variables, what each variable represents, and the constraint.
Solution:
Equation: 2x + 5y = 30
where x represents the number of pounds of peanuts and
y represents the number of pounds of walnut and
the constraint is that he has $28
Points
Description
2
The student correctly identifies the equation and the variables and constraints.
1
The student correctly identifies either the equation OR the variables and constraints.
0
The student does not correctly identify the equation NOR the variables and constraints.
11. Graph the solution to the inequality 3x + 4y ≤ 24.
Points
Description
2
The student correctly graphs the line and shades the correct side of the line.
1
The student either correctly graphs the line OR shades the correct side of the line.
0
The student does not correctly graph the line and shades the incorrect side of the line.
12. Given a system of four inequalities, explain the steps you would use to find the solution to the system. Your steps should include every detail necessary to arrive at a correct solution.
Student responses will vary, but should include the following details:
- Graphing each equation and using the appropriate style line (dashed or solid).
- Selecting test points to determine which side of each line to shade.
- Finding the region that is shaded by all four inequalities.
Points
Description
2
The student performs all three of the required activities correctly:
- graphing each equation and using the appropriate style line (dashed or solid).
- selecting test points to determine which side of each line to shade.
- finding the region that is shaded by all four inequalities.
1
The student performs one or two of the required activities correctly.
0
The student does not perform any of the required activities correctly.
Performance Assessment:
A factory makes tennis shoes and sandals. Tennis shoes take 5 minutes per pair to make, and sandals take 6 minutes per pair to make. Also, tennis shoes require 4 square feet of material per pair to make, and sandals require 2 square feet of material per pair to make. The factory must use less than 40 square feet of material per hour.
Write and solve a system of inequalities to represent all the possible combinations of tennis shoes and sandals the factory can make in 1 hour. Use x to represent the number of pairs of tennis shoes the factory makes, and y to represent the number of pairs of sandals the factory makes.
Performance Assessment Scoring Rubric:
Sample response:
5x + 6y ≤ 60
4x + 2y < 40
x ≥ 0
y ≥ 0
Points
Description
4
Student has all four inequalities correct, has graphed them all correctly (including dashed vs. solid lines), and has shaded them all correctly. The student has also clearly indicated where the solution region is.
3
The student has three of the four inequalities written and graphed correctly, or has them all correct except for dashed vs. solid lines, or fails to clearly indicate the solution region.
2
The student has two of the four inequalities written and graphed correctly.
1
The student has one of the four inequalities written and graphed correctly.
0
The student has none of the four inequalities written and graphed correctly and displays little or no understanding of the topic.
DRAFT 10/12/2011
