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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

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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

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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

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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)

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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

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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

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

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

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4) y + 2 0

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6) x + y < 0

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8) x + 2y 0

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10) x – y < 1

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

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12) x + 2y 2

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14) 2x – 3y < 6

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18) 2(y – 1) > 3(x + 1)

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

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

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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

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