1. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 2 Equations, Inequalities and Problem Solving

2. 22 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 1. Solve for y. 2. Solve for C. 3. Solve for I. 4. Solve for x. Bellwork:

3. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 2.8 Linear Inequalities and Problem Solving

4. 44 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

5. 55 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Linear Inequalities An inequality is a statement that contains one of the symbols:, ≤, or ≥. EquationsInequalities x = 3x > 3 12 = 7 – 3y12 ≤ 7 – 3y

6. 66 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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 {x  x  7} Represents the set {x  x > – 4} Graphing Solutions

7. 77 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Represents the interval (-∞, 7] Represents the interval (-4, ∞) 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 ] (

8. 88 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Interval Notation Set NotationInterval NotationGraph x < #(-∞, #) x ≤ #(-∞,#] x > #(#, ∞) x ≥ #[#, ∞) Parenthesis when the endpoint is not included Brackets when the endpoint is included Infinity always has a parenthesis ) ] [ ( smallest to LARGEST

9. 99 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Which is the graph of ? Example ) ) ] ] -2 5 5

10. 10 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Addition Property of Inequality If a, b, and c are real numbers, then a < banda + c < b + c are equivalent inequalities.

11. 11 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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. 2.If a, b, and c are real numbers, and c is negative, then a bc are equivalent inequalities. stays the same sign flips

12. 12 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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. Solving Linear Inequalities

13. 13 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set. 3x + 9 ≥ 5x – 5 Apply the distributive property. 3x – 3x + 9 ≥ 5x – 3x – 5 Subtract 3x from both sides. 9 ≥ 2x – 5 Simplify. 14 ≥ 2x Simplify. 7 ≥ x Divide both sides by 2. 9 + 5 ≥ 2x – 5 + 5 Add 5 to both sides. 3x + 9 ≥ 5(x – 1) Interval Notation (-∞, 7] Example [ 7

14. 14 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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. Divide both sides by 4. Example Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set. Interval Notation (-9/2, ∞) ) -9/2

15. 15 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Solve: 3x + 8 ≥ 5. Graph the solution set. Example ] Interval notation [-1,∞)

16. 16 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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

17. 17 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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. The number of people who can be invited must be 63 or less to stay within the budget.

18. 18 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: 1. What does interval notation represent? 2. What does a parenthesis mean? a bracket? 3. Name 3 words/phrases that mean “less than or equal to”