2.6 Linear Inequalities In two Variables Algebra 2

Definitions Linear Inequality: an inequality that can be Witten in one of the following forms 𝐴𝑥+𝐵𝑦<𝐶 𝐴𝑥+𝐵𝑦≤𝐶 𝐴𝑥+𝐵𝑦>𝐶 𝐴𝑥+𝐵𝑦≥𝐶 Solution of a linear inequality: The values of x and y that make the inequality true

Example Check whether the ordered pair is a solution of 4𝑥−2𝑦≥8. (3, 3) (-2, -9) Check whether the ordered pair is a solution of 3𝑥−𝑦≤6. (2, 0) (-1, 10)

Investigating the Graph of an Inequality Graph the following points (0, 0), (0, 2), (0, 4), (-2, 0), (-2, 2), (-2, 4), (-4, 0), (-4, 2), (- 4, 4), (-2, -2), (-4, -2), (-2, -4), (-4, -4), (0, - 2), (0, -4), (2, 0), (4, 0), (2, 2), (2, 4). (4, 2), (4, 4), (2, -2), (2, -4), (4, -2), and (-4, -4) Test each circled point to see whether it is a solution of 𝑥+𝑦≥1. If t is a solution, color it blue. If it is not a solution, color it red. Graph the line 𝑥+𝑦=1. What relationship do you see between the colored points and the line?

Investigating the Graph of an Inequality Describe a general strategy for graphing an inequality in two variables Graphing A Linear Inequality 1) Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for ≤ or ≥ 2) To decide which side of the boundary to shade, test a point no on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane

Examples Graph the following inequalities 𝑦≥2 𝑥>−1 4𝑥+2𝑦≥8 3𝑥−𝑦<3

Example You have $200 to spend on CDs and music videos. CDs cost $10 and music videos cost $15. Write a linear inequality in two variables to represent the number of CDs x and music videos y you can buy. Graph the inequality. Give three possible combinations of the number of CDs and music videos you can buy.

Example You have $15 for fresh fruit for a salad. Write a linear inequality in two variables for how many pound of strawberries x at $1.25 per pound and cherries y at $2.40 per pound you can buy. What are the intercepts of the graph?