Day 8 – Linear Inequalities

Jason collects card-game packages Jason collects card-game packages. The cost of each card packages for Game A is $3, and the cost of each card packages for Game B is $2. How many card packages for each game Jason buy? The answer to this question can be found using the graph of a linear inequality.

Graphing a Linear Inequality 1. Let 𝑥 represent the number of card packages for Game A, and let 𝑦 represent the number of card packages for Game B. write an inequality to represent the amount of money Jason can spend each week on card packages. 𝟑𝐱+𝟐𝐲≤𝟏𝟐 2. Replace the inequality symbol with an equal sign in the inequality you wrote in Step 1. Solve for y, and graph the equation on a coordinate plane. This line is called the boundary line of the related inequality.

3. The following table shows you can use substitution to find some points that satisfy inequality. Copy and complete the table.

4. Which points in the table make inequality true 4. Which points in the table make inequality true? Plot them on the graph you drew in Step 2. 𝟎, 𝟎 , 𝟏, 𝟏 , 𝟐, 𝟏 , 𝟐, 𝟐 , (𝟑, 𝟏) 5. Shade the region on the side of the line containing the points you plotted in Step 4. What is true about all of the points in this region? They all make the inequality 𝟑𝒙+𝟐𝒚≤𝟏𝟐 true. 6. Are all of the points in the shaded region solutions to the inequality? What are the reasonable domain and range for this real-world situation? Yes; 𝟎≤𝒙≤𝟒;𝟎≤𝒚≤𝟔

7. Can you list 𝑎𝑙𝑙 of the possible numbers of card packages from Games A and B That Jason can buy in one week? Explain. Why is the ordered pair 2 1 2 , 1 1 2 a solution to the inequality but not a real answer to the problem in Exploration 1? These coordinates produce a true statement when substituted in 𝟑𝒙+𝟐𝒚≤𝟏𝟐, but this is not a solution because you cannot buy partial packages.

A Greater- Than Inequality 8. What is the equation of the boundary line for the inequality 𝑦<2𝑥−4? 𝒚=𝟐𝒙−𝟒 9. Graph the boundary line. If an inequality contains the symbol < or >, the boundary line 𝑖𝑠 𝑛𝑜𝑡 part of the graph and is shown as a 𝑑𝑎𝑠ℎ𝑒𝑑 line. If the inequality contains the symbol ≤ or ≥, the boundary line is part of the graph and is shown as a 𝑠𝑜𝑙𝑖𝑑 line. Is the boundary line in the inequality 𝑦>2𝑥−4 solid or dashed?

9. Dashed

10. Copy and complete the following table to find points that make the inequality true. 11. Plot the points from the table that satisfy the inequality. Do the points lie above or below the boundary line? Shade the region containing these points.

10. Copy and complete the following table to find points that make the inequality true. 11. Plot the points from the table that satisfy the inequality. Do the points lie above or below the boundary line? Shade the region containing these points.

11.

12. Explain how to use the inequality to determine which side of the line to shade. Substitute the coordinates of different points into the inequality. The points that make the inequality true will all be on the side of the boundary line that is to be shaded.