Activities

1. List five coordinate points that are solutions to this inequality and five that are not solutions.

Solutions: (     ,     ), (     ,      ), (      ,      ), (       ,       ), (      ,      )

Not Solutions: (     ,     ), (     ,      ), (      ,      ), (       ,       ), (      ,      )

2. Determine the system of equations graphed below.

3. Graph the system of equations where y > 3 and x ≤ 2 on the coordinate plane.

4. Graph y < -5x – 1 and y > 4 + x on the coordinate plane below.

5. Graph this system of inequalities:  y > -2x – 1 and y > 2x + 1.

6. Find the solutions to this system of inequalities on the graph below.

5y + 3x ≤ 4(1 + y)       and      x – 2y > 6

7. Write the system of inequalities graphed here.

8. What is one of the infinite solutions to the system y > 10x – 1 and y ≤ 1 – 10x?

9. Fill in the correct inequality signs if the system of inequalities y ___ 7x + 1 and y ___ 4x -2 have (-1, -6) and (-3, -10) as solutions.

10. How many fewer solutions does the system y ≥ x + 2 and y ≤ x + 2 have than the system x > 2 and y > 2?

11. A system of inequalities is made up of y > 2x + a, and -2y ≥ -4x -2b. For which values of a and b does this system have no solutions?

12. Create a system of inequalities where the first inequality has a slope of -1 and the other has a slope of -3, be sure that the value (3, 3) is part of the solution to your system.

13. What is one value that is not a solution to the following system of equations:
    y > 4 – x
    y ≥ -1x + 4

14. How many solutions do the following systems of inequalities contain?

1. y > x + 3, y ≤ -2x + 7
2. y > 3x + 5, y > 3x + 7
3. y > x and y < x
4. x > 100, y ≤ 1000

15. Find the solutions to the system of three inequalities below.

y ≤ -x , y > 3x, and y > 2

16. What is one coordinate point that is a solution to y > 3 and y < 4?
    
    
17. The sum of two numbers is greater than 5 and the result when the second number is subtracted from the first is less than -3; graph the possible values for the two numbers.

18. Create a system of three inequalities whose solutions are all within the triangle show on the graph below.

19. Josiah knows where he should graph line L and line M on the coordinate plane below. However, he does not know if they should be dotted or solid lines. Also, he doesn’t know where to shade. If his friend, LaRynn tells him that the point (100, -100) is a solution, but the points (-2, 2), (-2, 3), (-3, 2) and (-2, 0) are not solutions. Determine if any of the lines should be dotted, and where Josiah should shade the graph.

20. Mr. James is a substitute teacher and he has never graphed a system of linear inequalities before, but he has been asked to explain them to a class later in the day. He kindly asks you to write down a process for graphing any system of inequalities that will help students later on. Think about the problem “Solve the system of inequalities y > 4x – 5 and y ≥ -3x + 2”, but do not use them in your explanation.

Answer Key/Rubric

1. Various answers including:
   Solutions: (0, 0), (0, 1), (-1, 0), (1, 1), (-5, -5)
   Not Solutions: (0, 2), (2, 0), (0, -3), (10, 10), (5, -5)

2. and
   
   
3. 
   
4. 
   
5. 
   
6. 
   

7. x < 2,

8. (-10, 0)

9. y ≥ 7x + 1 and y ≥ 4x -2

10. None, they both have infinite solutions

11. When a > b

12. Various answers including:  y < -1x + 20 and y < -3x + 15

13. Various answers including answers that are on or below the line y = -x + 4, example: (1, 3)

14. How many solutions do the following systems of inequalities contain?

1. Infinite solutions
2. Infinite solutions
3. No solutions
4. Infinite solutions

16. Various solutions including (3, 3.5)

18. y ≥ 0, x ≥ 0, y ≤ -x + 8

19. Line M should be dotted, Line L should be solid, The area directly to the right of the intersection should be shaded

20. Various answers including:

Step 1: Solve both inequalities for the y variable.

Step 2: Starting with the y-intercept, graph each line using the slope. If the line has a > or < sign, draw it with a dotted line. If the inequality sign is ≤ or ≥ use a regular solid line.

Step 3: Determine where the solutions of the first inequality are located. If the sign is ≥ or > then you shade above that line lightly. If the sign is < or ≤ then shade lightly below that line.

Step 4: Determine where the solutions of the 2^nd inequality are located the using the same manner as the first.

Step 5: The solutions are found in the area where both shading overlaps. Erase any shading outside of that area and darken in the overlapping area.