1. Section 2.8 and 3.3 Graphing Inequalities
Monday November 10 and Wednesday November 12

2. Think about it…
Using the table below, fill in the first two columns. First you will write anything you know about graphing linear equations. In the second column, you will predict what will happen when graphing an inequality (ex: 𝑦<3𝑥+2)
What I know about graphing lines…
What I predict will happen when graphing inequalities…
What actually happened when I graphed inequalities… *

3. Explore: Graphing Inequalities
You will receive a two-sided worksheet. One side will say 2.8 and the other will say 3.3. We will be working on section 2.8 for right now.
You may work with your partner for this activity.
Make sure you use a straight edge for drawing your lines.

4. How did we do?
𝑦<2𝑥+3
Graph the inequality as an equation. What do you think would be considered less than that line?
Now, shade the area of the graph that is less than the line you drew.
Would the line you drew be included with the shaded area? How do you know?
If your line is included make sure it is a solid line. If it is not included with the shaded area, then make sure your line is a dotted line.

5. How did we do?
𝑦≥ 1 3 𝑥+2
Graph the inequality as an equation. What do you think would be considered greater than or equal to that line?
Now, shade the area of the graph that is greater than or equal to the line you drew.
Would the line you drew be included with the shaded area? How do you know?
If your line is included make sure it is a solid line. If it is not included with the shaded area, then make sure your line is a dotted line.

6. Explanation
A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane bounded by a line.
The line is the boundary of the graph.
The boundary separates the coordinate plane into two half-planes, one of which consists of solutions to the inequality.
To determine which half-plane to shade, pick a test point that is not on the boundary and check whether that point satisfies the inequality.
If it does, shade the half-plane which includes the test point. If it doesn’t, shade the other half-plane.
Tip: Test the origin if it does not fall on the boundary.

7. Graphing an Absolute Value Inequality
You Try: 𝑦> 1 2 𝑥+3 −4 a= h= k=
Example: 𝑦≤2 𝑥−1 +3 a= h= k=

8. Let’s work backwards: Writing an inequality based on a graph
Example:
What is the y-intercept of the line?
What is another point of the line?
What is the slope of the line?
What is the equation of the line?
Is the shading including the line or not?

9. Let’s work backwards: Writing an inequality based on a graph
You Try:
What is the y-intercept of the line?
What is another point of the line?
What is the slope of the line?
What is the equation of the line?
Is the shading including the line or not?

10. What actually happened?
Refer back to your chart from our first activity. What actually happened when we graphed systems of inequalities. How is the same or different from what you expected would happen?
What I know about graphing lines…
What I predict will happen when graphing inequalities…
What actually happened when I graphed inequalities… *

11. 3.3 System of Inequalities Explore: Graphing Systems of Inequalities
Take out the two-sided worksheet you received earlier. Now, we will be working on section 3.3.
You may work with your partner for this activity.
Make sure you use a straight edge for drawing your lines.

12. How did we do?
1. 𝑥−𝑦≥1
2𝑥+3𝑦≤21
Graph the first inequality and shade the solution with one color.
Graph the second inequality and shade the solution with a second color.
Which are would represent the solution to the system of inequalities?

13. How did we do? 2. 𝑦>𝑥 𝑦<𝑥+2
Graph the first inequality and shade the solution with one color.
Graph the second inequality and shade the solution with a second color.
Which are would represent the solution to the system of inequalities?

14. Solution of a system of inequalities
You can solve a system of inequalities in more than one way, but graphing the solution is usually the most appropriate
If you graph each inequality, the overlapping region is the solution of the system.
To check your answer, choose a point in the region and test it in both inequalities.

15. Using a system of inequalities
Your city’s cultural center is sponsoring a concert to raise at least $30,000 for the city’s Youth Services. Tickets are $20 for balcony seats and $30 for orchestra seats. If the center has 500 orchestra seats, how many of each type of seat must be sold?

16. ROPES Warm-Up
Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that day, you bought six oranges and three grapefruits for a total of $4.80. What is the price of each type of fruit?

17. ROPES
Suppose you bought 10 oranges and 3 grapefruit for a total of $ Later that day, you bought six oranges and four grapefruits for a total of $9.30. What is the price of each type of fruit?

18. Evaluate
Graph the following system of inequalities and state whether the following points are solutions or not.
(0, 2) b) (1, 0) c) (3, 2)
𝑦≤2𝑥+3
𝑦≥− 1 2 𝑥+1