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Objective The student will be able to: solve inequalities. MFCR Lesson 1-7 9-25-14

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9-25-14 Bellwork

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Section 1.7—Linear Inequalities in One Variable Copy Key concepts from p. 94 Read examples 1 & 2

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1.7—Linear Inequalities in One Variable A linear inequality in one variable can be written in the form ax + b < 0, ax + b> 0, ax + b ≤ 0, or ax + b ≥ 0

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Properties of Inequalities 1.If a < b, then a + c < b + c. 2.If a < b, then a – c < b – c. 3.If c is positive and a < b, then ac < bc and a/c < b/c. 4.If c is negative and a bc and a/c > b/c. *Properties 3 and 4 indicate that if we multiply or divide an inequality by a negative value, the direction of the inequality sign must be reversed.

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1) Solve 5m - 8 > 12 + 8 + 8 5m > 20 5 5 m > 4 5(4) – 8 = 12 1.Draw “the river” 2.Add 8 to both sides 3.Simplify 4.Divide both sides by 5 5.Simplify 6.Check your answer 7.Graph the solution o 453

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2) Solve 12 - 3a > 18 - 12 - 12 -3a > 6 -3 -3 a < -2 12 - 3(-2) = 18 1.Draw “the river” 2.Subtract 12 from both sides 3.Simplify 4.Divide both sides by -3 5.Simplify (Switch the inequality!) 6.Check your answer 7.Graph the solution o -2-3

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Which graph shows the solution to 2x - 10 ≥ 4? 1.. 2. 3. 4. Answer Now

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3) Solve 5m - 4 < 2m + 11 -2m 3m - 4 < 11 + 4 + 4 3m < 15 3 3 m < 5 5(5) – 4 = 2(5) + 11 1.Draw “the river” 2.Subtract 2m from both sides 3.Simplify 4.Add 4 to both sides 5.Simplify 6.Divide both sides by 3 7.Simplify 8.Check your answer 9.Graph the solution o 564

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4) Solve 2r - 18 ≤ 5r + 3 -2r -18 ≤ 3r + 3 - 3 - 3 -21 ≤ 3r 3 3 -7 ≤ r or r ≥ -7 2(-7) – 18 = 5(-7) + 3 1.Draw “the river” 2.Subtract 2r from both sides 3.Simplify 4.Subtract 3 from both sides 5.Simplify 6.Divide both sides by 3 7.Simplify 8.Check your answer 9.Graph the solution ● -7-6-8

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6) Solve -2x + 6 ≥ 3x - 4 1.x ≥ -2 2.x ≤ -2 3.x ≥ 2 4.x ≤ 2 Answer Now

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5) Solve 26p - 20 > 14p + 64 -14p 12p – 20 > 64 + 20 + 20 12p > 84 12 12 p > 7 26(7) – 20 = 14(7) + 64 1.Draw “the river” 2.Subtract 14p from both sides 3.Simplify 4.Add 20 to both sides 5.Simplify 6.Divide both sides by 12 7.Simplify 8.Check your answer 9.Graph the solution o 786

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What are the values of x if 3(x + 4) - 5(x - 1) < 5? Answer Now 1.x < -6 2.x > -6 3.x < 6 4.x > 6

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Objectives The student will be able to: 1. solve compound inequalities. 2. graph the solution sets of compound inequalities.

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Compound Inequalities To solve a compound inequality, isolate the variable x in the “middle.” The operations performed on the middle portion of the inequality must also be performed on the left-hand side and right-hand side. Ex. 1: -2 ≤ 3x + 1 < 5 Ex. 2: -8 < 5x – 3 ≤ 12

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What is the difference between and and or ? AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A AB B

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1) Graph x < 4 and x ≥ 2 342 ● a) Graph x < 4 b) Graph x ≥ 2 342 o c) Combine the graphs ● 342 o d) Where do they intersect? ● 342 o

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2) Graph x < 2 or x ≥ 4 342 ● a) Graph x < 2 b) Graph x ≥ 4 342 o c) Combine the graphs 342 o 342 ●

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3) Which inequalities describe the following graph? -2-3 o o Answer Now 1.y > -3 or y < -1 2.y > -3 and y < -1 3.y ≤ -3 or y ≥ -1 4.y ≥ -3 and y ≤ -1

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When written this way, it is the same thing as 6 < m AND m < 8 It can be rewritten as m > 6 and m < 8 and graphed as previously shown, however, it is easier to graph everything between 6 and 8! 4) Graph the compound inequality 6 < m < 8 786 o o

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5) Which is equivalent to -3 < y < 5? 1.y > -3 or y < 5 2.y > -3 and y < 5 3.y 5 4.y 5 Answer Now

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6) Which is equivalent to x > -5 and x ≤ 1? 1.-5 < x ≤ 1 2.-5 > x ≥ 1 3.-5 > x ≤ 1 4.-5 < x ≥ 1 Answer Now

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7) 2x < -6 and 3x ≥ 12 1.Solve each inequality for x 2.Graph each inequality 3.Combine the graphs 4.Where do they intersect? 5.They do not! x cannot be greater than or equal to 4 and less than -3 No Solution!! -30-6 o -30-6 o 471 o ● 471 o ●

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8) Graph 3 < 2m – 1 < 9 Remember, when written like this, it is an AND problem! 3 < 2m – 1 AND 2m – 1 < 9 Solve each inequality. Graph the intersection of 2 < m and m < 5. 05 -5

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9) Graph x < 2 or x ≥ 4 05 -5

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10) Graph x ≥ -1 or x ≤ 3 The whole line is shaded!!

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Practice Problems

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Solving a Compound Inequality Application Beth received grades of 87%, 82%, 96%, and 79% on her last four algebra tests. To graduate with honors, she needs at least a B in the course. a)What grade does she need to make on the 5 th test to get a B in the course? Assume that the tests are weighted equally and that to earn a B the average of the test grades must be at least 80% but less than 90%. b)Is it possible for Beth to earn an A in the course if an A requires an average of 90% or more?

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Solving a Linear Inequality Application The number of registered passenger cars, N (in millions), in the U.S. has risen since 1960 according to the equation N = 2.5t + 64.4, where t represents the number of years after 1960 (t = 0 corresponds to 1960, t = 1 corresponds to 1961, etc.) a)For what years was the number of registered passenger cars less than 89.4 million? b)For what years was the number of registered passenger cars between 94.4 million and 101.9 million?

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Exit Ticket – Hand in before leaving class. a. What are the solutions to -2 < x + 2 ≤ 5 b. Graph the solutions to the above compound inequality

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