We use inequalities when there is a range of possible answers for a situation. I have to be there in less than 5 minutes, This team needs to score at least a goal to have a chance of winning, To get into the city, I need at most $6.50 for train fare are all examples of situations where a limit is specified, but … I < 5 minutes T 1 goal F $6.50 … a range of possibilities exist beyond that limit. Thats what we are interested in when we study inequalities. linear inequalities in two variables 3.7

Sara and Ali want to donate some money to a local food pantry. To raise funds, they are selling necklaces and earrings that they have made themselves. Necklaces cost $8 and earrings cost $5. What is the range of possible sales they could make in order to donate at least $100? amount of money earned from selling earrings + amount of money earned from selling necklaces $100 5y + 8x $100 … The objective of todays lesson is to graph such linear inequalities on the coordinate plane, and determine a solution set

Example 1 Graph each inequality x + 2y Step1: Put the inequality in slope intercept form: y = mx + b and graph it. 2y -x + 4 b Caution or (solid) (dashed) IF Step2 Select a testing point thats not on the line, substitute it into the inequality. If the result is true, shade that region. Otherwise shade the opposite one (0) True … Test (0, 0).. The easiest point

Example 2 3x 2(y – 1) 3x 2y – 2 b -2y -3x – 2 Caution or (solid) (dashed) IF -2 Test (0, 0) 0 -2 true … 3(0) 2(0 – 1) 3x 2(y – 1)

Example 3 x + 1 < 0 x < – 1 Vertical line through x = – 1 Test (0, 0) False … … so we shade the region that does not contain point (0, 0) x < – 1 0 < – 1

Example 4 y – 2 > 0 y > 2 horizontal line through y = 2 Test (0, 0) False … … so we shade the region that does not contain point (0, 0) y > 2 0 > 2

Example 5 x + y > 0 y > -x + 0 b Test (-1, -1) False … -1 + (-1) > 0 -2 > 0

Written Exercises.. page 138 2) x – 1 < 0

4) y + 2 0

6) x + y < 0

8) x + 2y 0

10) x – y < 1

Page 138 #s 12, 14, 16, 18 HomeworkHomework

12) x + 2y 2

14) 2x – 3y < 6

16)

18) 2(y – 1) > 3(x + 1)

Graph each system of inequalities b Test (0, 0) True … b Test (0, 0) True …

Page 138 #s 20, 24, 28, 32 HomeworkHomework

Graph each system of inequalities b Test (0, 0) True … 0 < < 2 Test (0, 0) False … 0 > 1 23) 1 – y < 0 – y < – 1 y > 1 Horizontal line

20)

24)

28)

32)