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Chapter 2 Equations, Inequalities and Problem Solving Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: A C 1. Solve Ax By C for y. y x B B 9 5 160 2. Solve F C 32 for C. C F 5 9 9 A I 3. Solve Q for I. I A QL L 4. SolveY a m(x b) for x. Y a b x m Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 2 2.8 Linear Inequalities and Problem Solving Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Objectives: Graph inequalities on a number line Use the addition/multiplication property to solve inequalities Solve problems modeled by inequalities Write solutions in interval notation Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 4 Linear Inequalities An inequality is a statement that contains one of the symbols: < , >, ≤ , or ≥. Equations x=3 Inequalities x>3 12 = 7 – 3y 12 ≤ 7 – 3y Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 5 Graphing Solutions Graphing solutions to linear inequalities in one variable • Use a number line • Use a bracket (closed circle) at the endpoint of a interval if you want to include the point • Use a parenthesis (open circle) at the endpoint if you DO NOT want to include the point Represents the set {xx 7} Represents the set {xx > – 4} Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 6 Graphing Solutions Graphing solutions to linear inequalities in one variable • Use a number line • Use a bracket (closed circle) at the endpoint of a interval if you want to include the point • Use a parenthesis (open circle) at the endpoint if you DO NOT want to include the point ] ( Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Represents the interval (-∞, 7] Represents the interval (-4, ∞) 7 Interval Notation Set Notation Interval Notation Graph x<# (-∞, #) x≤# (-∞,#] x># (#, ∞) x≥# [#, ∞) ) ] ( [ smallest to LARGEST •Parenthesis when the endpoint is not included •Brackets when the endpoint is included •Infinity always has a parenthesis Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 8 Example Which is the graph of 2 x 5 ? ] ) -2 5 ) ] -2 5 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 9 Addition Property of Inequality If a, b, and c are real numbers, then a < band a + c < b + c are equivalent inequalities. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 10 Multiplication Property of Inequality 1. If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities. stays the same 2. If a, b, and c are real numbers, and c is negative, then a < b and ac > bc are equivalent inequalities. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. sign flips 11 Solving Linear Inequalities To Solve Linear Inequalities in One Variable Step 1: If an inequality contains fractions, multiply both sides by the LCD to clear the inequality of fractions. Step 2: Use distributive property to remove parentheses if they appear. Step 3: Simplify each side of inequality by combining like terms. Step 4: Get all variable terms on one side and all numbers on the other side by using the addition property of inequality. Step 5: Get the variable alone by using the multiplication property of inequality. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 12 Example Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set. 3x + 9 ≥ 5(x – 1) 3x + 9 ≥ 5x – 5 3x – 3x + 9 ≥ 5x – 3x – 5 9 ≥ 2x – 5 9 + 5 ≥ 2x – 5 + 5 14 ≥ 2x 7≥x Apply the distributive property. Subtract 3x from both sides. Simplify. Add 5 to both sides. Simplify. Divide both sides by 2. Interval Notation (-∞, 7] [ 7 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 13 Example Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set. 7(x – 2) + x > –4(5 – x) – 12 7x – 14 + x > –20 + 4x – 12 Apply the distributive property. 8x – 14 > 4x – 32 Combine like terms. 8x – 4x – 14 > 4x – 4x – 32 Subtract 4x from both sides. 4x – 14 > –32 Simplify. 4x – 14 + 14 > –32 + 14 Add 14 to both sides. 4x > –18 Simplify. Interval 9 x Divide both sides by 4. Notation 2 (-9/2, ∞) ) -9/2 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 14 Example Solve: 3x + 8 ≥ 5. Graph the solution set. 3x 8 5 3x 8 8 5 8 3 x 3 3x 3 3 3 x 1 Interval notation [-1,∞) ] -1 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 15 Example You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget. Let x represent the number of people Set up fee + cost per person × number of people ≤ 1200 250 + 15x ≤ 1200 continued Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 16 continued You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget. 250 15x 1200 15 x 950 15 x 950 15 15 x 63.3 The number of people who can be invited must be 63 or less to stay within the budget. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 17 Closure: 1. 2. 3. What does interval notation represent? What does a parenthesis mean? a bracket? Name 3 words/phrases that mean “less than or equal to” Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 18