1. Algebra 2 Solving Inequalities Lesson 1-5 Part 1

2. Goals Goal To solve and graph inequalities. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Essential Question Big Idea: Solving Inequalities How do you solve inequalities? –Students will apply the Properties of Inequality to solve an inequality. –Students will find all of the values of a variable that make an inequality true.

4. Vocabulary None

5. What is an Inequality? An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: Inequalities work like equations, but they tell you whether one expression is bigger or smaller than the expression on the other side. < “is less than” > “is greater than” ≤ “is less than or equal to” ≥ “is greater than or equal to” An inequality is a mathematical sentence that states that two expressions are not equal.

6. Graphing Inequalities x < c When x is less than a constant, you darken in the part of the number line that is to the left of the constant. Also, because there is no equal line, we are not including where x is equal to the constant. That means we are not including the endpoint. One way to notate that is to use an open hole at that point. x > c When x is greater than a constant, you darken in the part of the number line that is to the right of the constant. Also, because there is no equal line, we are not including where x is equal to the constant. That means we are not including the endpoint. One way to notate that is to use an open hole at that point.

7. x < c When x is less than or equal to a constant, you darken in the part of the number line that is to the left of the constant. Also, because there is an equal line, we are including where x is equal to the constant. That means we are including the endpoint. One way to notate that is to use an closed hole at that point. x > c x > c When x is greater than or equal to a constant, you darken in the part of the number line that is to the right of the constant. Also, because there is an equal line, we are including where x is equal to the constant. That means we are including the endpoint. One way to notate that is to use a closed hole at that point. Graphing Inequalities

8. Graph each of these inequalities. Your Turn: Graphing Inequalities 1) l ≤ 3 2) m < –2 3) j ≥ 10 4) 3 ≥ y 5) k > 6 6)10 ≤ x 7)j > –5 8. State the inequality represented on the number line below. x ≥ –1 k > –7 k ≤ 2

9. Writing and Graphing Inequalities

10. <> ≤≥ less than fewer than greater than more than exceeds less than or equal to no more than at most greater than or equal to no less than at least Inequality symbols "Solving'' an inequality means finding all of its solutions. A "solution'' of an inequality is a number which when substituted for the variable makes the inequality a true statement.

11. Anthony is shopping for a birthday gift for his cousin Robert. He has $25 dollars in his wallet. Write an inequality that shows how many dollars he can spend on the gift. Teresa is only allowed to swim outside if the temperature outside is at least 85 °F. Write an inequality that shows the temperature in degrees Fahrenheit at which Teresa is allowed to swim. t ≥ 85 In order to achieve an ‘A’ in math, Ivy needs to score more than 95% on her next test. Write an inequality that shows the test score Ivy needs to achieve in order to earn her ‘A’ in math. i > 95 a ≤ 25 Writing Inequalities

12. Write an inequality for each situation. A. There are at least 35 people in the gym. p ≥ 35 B. The carton holds at most 12 eggs. e ≤ 12 “At least” means greater than or equal to. “At most” means less than or equal to. Let p = the number of people in the gym. Let e = the number of eggs the carton hold. Example:

13. Write an inequality for each situation. A. There are at most 10 gallons of gas in the tank. g ≤ 10 B. There are fewer than 10 yards of fabric left. y < 10 “At most” means less than or equal to. “Fewer than” means less than. Let g = the number of gallons of gas. Let y = the yards of fabric. Your Turn:

14. Write an inequality for each statement. A. A number m multiplied by 5 is less than 25. 5m < 25 B. The sum of a number y and 16 is no more than 100. y + 16 ≤ 100 m  5 < 25 The sum of a number y and 16 is no more than 100 A number m multiplied by 5 is less than 25. y + 16 ≤ 100 Example:

15. Write an inequality for each statement. A. A number y plus 14 is greater than 21. y + 14 > 21 B. A number t increased by 7 is more than 11 t + 7 > 11 y + 14 > 21 A number t is increased by 7 is more than 11 A number y plus 14 is greater than 21. t + 7 > 11 Your Turn:

16. Solving Inequalities You solve an inequality the same as an equation, with one important difference. If you multiply or divide both sides by a negative number, you must reverse the inequality symbol. The graph of an inequality is the solution set, the set of all points on the number line that satisfy the inequality.

17. Addition/Subtraction Property for Inequalities If a < b, then a + c < b + c If a < b, then a - c < b – c In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality. Addition/Subtraction Property for Inequalities If a < b, then a + c < b + c If a < b, then a - c < b – c In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality. Ex. A Solve and graph the solution of x – 2 > 5 on a number line. Solve: x – 2 > 5 +2 +2 Using addition property of inequalities x > 7 Graph: –707 Ex. B Solve and then graph the solution of 3x ≤ 6 + 2x on a number line. Solve: 3x ≤ 6 + 2x -2x Subtraction property of inequalities x ≤ 6 Graph: –11402356

18. Your Turn: Solve and graph each inequality. 1)x + 2 < 8 x < 6 2) j + 2 ≥ –3 j ³ –5 3) 4x – 3 < 3x x < 3 4) 2t + 1 ≤ t – 10 t £ –11 5) 6(x – 1) ≥ 5(x + 2) x ³ 16

19. Multiplication/Division Properties for Inequalities with NEGATIVE Numbers Given real numbers a, b, and c, if a > b and c < 0 then ac < bc. Given real numbers a, b, and c, if a > b and c < 0 then < In other words, multiplying or dividing the same NEGATIVE number to both sides of an inequality REVERSES the direction of the inequality, otherwise the inequality statement will be false. Multiplication/Division Properties for Inequalities with NEGATIVE Numbers Given real numbers a, b, and c, if a > b and c < 0 then ac < bc. Given real numbers a, b, and c, if a > b and c < 0 then < In other words, multiplying or dividing the same NEGATIVE number to both sides of an inequality REVERSES the direction of the inequality, otherwise the inequality statement will be false. a c b c Remember — the sign needs to change. Solve: Solution: Solve: Solution: x < –4 Remember — the sign needs to change. Solve: Solution: Solve: Solution: y ≥ - 3

20. Solve Solution Subtract 5. Divide by – 3; reverse direction of the inequality symbol when multiplying or dividing by a negative number. Don’t forget to reverse the symbol here. Example:

21. 1)6) 2(x – 3) – 3(2 – x) > 8 2)7) –4(3 – 2x) > 5x + 9 3)8) 7 – 2(m – 4) £ 2m + 11 4)9) 3(2x + 6) – 5(x + 8) £ 2x – 22 x > –12 5) 1)6) 2(x – 3) – 3(2 – x) > 8 2)7) –4(3 – 2x) > 5x + 9 3)8) 7 – 2(m – 4) £ 2m + 11 4)9) 3(2x + 6) – 5(x + 8) £ 2x – 22 x > –12 5) x > 4 x > 7 m ³ 1 Your Turn:

22. To simplify and therefore solve an inequality in one variable such as x, you need to isolate the terms in x on one side and isolate the numbers on the other. Solving Inequalities 1. Multiply both sides by the LCD to clear all fractions. 2. Simplify grouping using the distributive property. 3. Simplify both sides by combining like terms. 4. Move all variables to one side. 5. Solve for the variable by conducting inverse operations in the reverse order. Don’t Forget: Multiplying or dividing the same NEGATIVE number to both sides of an inequality REVERSES the direction of the inequality. Multistep Inequalities

23. Solve 4 – 3x ≤ 7 + 2x. Graph the solution. Solution Subtract 4. Subtract 2x. Example:

24. Solution Divide by –5; reverse the direction of the inequality symbol. Example: Continued

25. 1) 2) 3) 4) 5) 1) 2) 3) 4) 5) 6x – 2 ≤ 4(x + 5)x ≤ 11 5x + 1 > 3(x + 3)x > 4 4(x + 1) 6 > 2xx < 1/2 8(x – 1) ≥ 4x – 4x ≥ 1 7(a – 4) 2 ≤ 4a a ≥ -28 Your Turn:

26. Special Cases Just like with equations, there are instances where an inequality will have No Solutions or All Real Numbers will be solutions of the inequality. In simplified form neither inequality contains variables. –If the variable divides out leaving a TRUE statement (7 > 4), then ALL REAL NUMBERS are solutions. –If the variable divides out leaving a FALSE statement (1 < -4), then there are NO SOLUTIONS.

27. Example: 4(x – 3) < 2(2x – 5) 4x – 12 < 4x – 10 – 12 < – 10 Solution: All Real Numbers True Statement Solve:

28. Example: 2(3x + 5) > 6(x – 1) + 17 6x + 10 > 6x – 6 + 17 6x + 10 > 6x + 11 10 > 11 NO SOLUTION Solve: False Statement

29. Your Turn: Solve: 5(t-3) ≥ 7-(t+4)+6t -15 ≥ 3 False Statement No Solution Solve: 9x-8<4(x+3)+5x -8 < 12 True Statement All Real Numbers

30. Steps for Solving Inequality Word Problems Step 1: Identify What You Are Looking For. Step 2: Give Names to the Unknowns. Step 3: Translate into the Language of Mathematics. Step 4: Solve the Inequality Found in Step 3. Step 5: Check the Reasonableness of Your Answer. Step 6: Answer the Question.

31. Example: Laura has $5.30 to spend on her lunch. She wants to buy a chicken salad costing $4.20 and decides to spend the rest on fruit. Each piece of fruit costs 45¢. Write an inequality to represent this situation, and then solve it to find how many pieces of fruit Laura can buy.

32. Solution: 1.Identify : How many pieces of fruit can Laura Buy. 2.Name : Let f = the number of fruit. 3.Translate : The cost of the chicken salad plus the cost of the fruit (cost per piece times the number of pieces) is less then or equal to the total amount she has to spend. 4.20 +.45f ≤ 5.30 Laura has $5.30 to spend on her lunch. She wants to buy a chicken salad costing $4.20 and decides to spend the rest on fruit. Each piece of fruit costs 45¢. Write an inequality to represent this situation, and then solve it to find how many pieces of fruit Laura can buy.

33. Solution: 4.Solve : 4.20 +.45f ≤ 5.30.45f ≤ 1.10 f ≤ 2.44 5.Check : The number of fruit (f) must be a whole number, therefore f = 2. And 4.20 +.45(2) ≤ 5.30 → 5.10 ≤ 5.30 checks. 6.Answer the Question : Laura can buy 2 pieces of fruit. Laura has $5.30 to spend on her lunch. She wants to buy a chicken salad costing $4.20 and decides to spend the rest on fruit. Each piece of fruit costs 45¢. Write an inequality to represent this situation, and then solve it to find how many pieces of fruit Laura can buy.

34. Your Turn: Javier has at most $15.00 to spend today. He buys a bag of pretzels and a bottle of juice for $1.59. If gasoline at this store costs $2.89 per gallon, how many gallons of gasoline, to the nearest tenth of a gallon, can Javier buy for his car?

35. Solution: 2. Name:Let g = the number of gallons of gasoline that Javier buys. The total cost of the gasoline is 2.89g. The cost of the pretzels and juice plus the total cost of the gasoline must be less than or equal to $15.00. Javier has at most $15.00 to spend today. He buys a bag of pretzels and a bottle of juice for $1.59. If gasoline at this store costs $2.89 per gallon, how many gallons of gasoline, to the nearest tenth of a gallon, can Javier buy for his car? 1. Identify: Find the number of gallons, to the nearest tenth, that Javier can buy.

36. Original inequality Subtract 1.59 from each side. Simplify. The cost of pretzels & juice plus the cost of gasoline is less than or equal to $15.00. 1.59+2.89g  15.00 3. Translate: 4. Solve: Solution:

37. 6. Answer the Question:Javier can buy up to 4.6 gallons of gasoline for his car. 5. Check:Since is actually greater than 4.6, Javier will have enough money if he gets no more than 4.6 gallons of gasoline.

38. Your Turn: Audrey is selling magazine subscriptions to raise money for the school library. The library will get $2.50 for every magazine subscription she sells. Audrey wants to raise at least $250 for the library. Write and solve an inequality to represent the number of magazine subscriptions, x, Audrey needs to sell to reach her goal. Solution: 2.50x ≥ 250 x ≥ 100 Audrey must sell at least 100 magazine subscriptions

39. Assignment Section 1-5 part 1, Pg 37 – 38; #1 – 6 all, 8 – 30 even, 34