Inequalities

1. Inequalities Algebraic Concepts and Applications

2. Overview • Graphing and Writing Simple Inequalities • Solving Simple Inequalities • Graphing and Writing Compound Inequalities • Solving Compound Inequalities • Solving Problems with Inequalities • The Average Problem • Money Application Inequalities Unit Content

3. Inequalities • Inequalities are similar to equations in that they show a relationship between two expressions. • We solve and graph inequalities in a similar way to equations. However, there are some differences that we will talk about later. • The main difference is that for linear inequalities the answer is an interval of values whereas for a linear equation the answer is most often just one value.

4. Inequalities • When writing inequalities, we use the following: > “greater than” < “less than” ≥ “greater than or equal to” ≤ “less than or equal to”

5. Graphing Inequalities of One Variable on a Number Line • When graphing inequalities on a number line, remember the following: • If the inequality sign is > or <, use an open circle • If the inequality sign is ≥ or ≤, use a closed circle

6. Graphing Inequalities of One Variable on a Number Line • Example: y > 3 • Example: y < 1

7. Graphing Inequalities of One Variable on a Number Line • Example: x ≥ -2 • Example: -4 ≥ z

8. Graphing Inequalities of One Variable on a Number Line Your turn! • x > -4 • 3 > y • z ≤ 0 • -2 ≤ x • 0 ≥ y

9. Graphing Inequalities of One Variable on a Number Line Write an inequality to match the following number lines:

10. Graphing Inequalities of One Variable on a Number Line Write an inequality to match the following number lines:

11. Writing Inequalities of One Variable on a Number Line • Expressions can often be expressed using inequalities. • Express the following situations using an inequality and show its graph: • Example: “In order to ride the roller coaster, you must be at least 48 inches tall.”

12. Writing Inequalities of One Variable on a Number Line • Example: “Mary needs at least forty more paper sales in order to meet her quota for the month.” • Example: “The speed limit on the interstate is 65 miles per hour.”

13. Writing Inequalities of One Variable on a Number Line • Example: “You must be younger than 3 years old to get free admission at the Montgomery Zoo.”

14. Writing Inequalities of One Variable on a Number Line Your Turn! • Miguel can spend no more than $100 on shoes. • Jessica has to spend at least $40 on her order to get free shipping • The secret number is less than eleven. • Robert’s age is more than twenty.

15. Solving Inequalities Lesson 2

16. Solving Inequalities • To solve an inequality, follow the same steps that you would when solving any equation. • Keep in mind that if you multiply or divide by a negative number in order to solve the inequality, you must flip the inequality sign!!

17. Solving Inequalities • Recall the steps for solving equations: • If there are parentheses, you must distribute first. • Next, ensure that both sides of the equation are completely simplified. (If there are like terms on the left, combine them. Likewise on the right.) • Next, ensure that your variables are on the same side of the equation. If they are not, add or subtract as needed to get them on the same side. • After following 1-3, you should have a two-step equation left to solve.

18. Solving Inequalities Solve the following inequalities: • Example: -2x ≥ 16

19. Solving Inequalities Solve the following inequalities: • Example: 6x – 5 < 10

20. Solving Inequalities Solve the following inequalities: • Example: -9x < -5x – 15

21. Solving Inequalities

22. Solving Inequalities

23. Solving Inequalities

24. Solving Inequalities

25. Graphing and Writing Compound Inequalities Lesson 3

26. Compound Inequalities

27. Graphing Compound Inequalities

28. Graphing Compound Inequalities

29. Graphing Compound Inequalities

30. Graphing Compound Inequalities

31. Graphing Compound Inequalities • Write a compound inequality that represents each situation: • “You must be between the ages of 18 and 55 to participate.” • “The interstate’s speed limit is 65mph but the minimum speed is 40mph.” • “The secret number is either more than eleven or less than five.”

32. Solving Compound Inequalities Lesson 4

33. Solving Compound Inequalities

34. Solving Compound Inequalities

35. Solving Compound Inequalities

36. Solving Compound Inequalities

37. Solving Compound Inequalities

38. Applications of Inequalities The Average Problem Lesson 5

39. The Average Problem • How do you take an average? Sum all of the numbers and divide by how many there are. Example: Find the average of the following scores: 80, 62, 95, 70, 73, 100

40. The Average Problem • John wants to have at least an 80% test average in his math class. His test grades so far are: 90, 80, 70, 73, and 85. What is the lowest he can make on his next test and still have an 80% test average?

41. The Average Problem • Averages aren’t always so simple…in your classes often your tests count more towards your grade than your homework or quizzes. • Example: Find Jed’s average if his class work counts 35% and his tests count 65%. His class work average is a 85 and his test average is a 70.

42. The Average Problem • From the last problem, can we write an equation that will show the process to average Jed’s grades? • Remember that his class work counts 35% and his tests count 65%.

43. The Average Problem • Jed wants to maintain a 75 average in his class. His past test grades were 60, 80, 73, and 67. Recall that his class work average was an 85. What is the minimum he can make on his next test and maintain a 75 overall average? • (Class work avg)(35%)+(Test Avg)(65%) = Final Avg.

44. The Average Problem • Jimmy wants to obtain a batting average of .300. Currently, he has hit 15 times out of 75 attempts. If Jimmy has luck and hits every time he goes to the plate from this moment, how many times will he have to hit the ball to bring his batting average up to .300?

45. Applications of Inequalities Money Applications Lesson 6

46. Money Applications • Tips for solving word problems: • Read the entire problem. • Highlight important key words • Identify variables • Be sure to answer the question being asked. • Double-check to ensure that your answer makes sense to answer your question. • Inequality Key Words • “at least” – means greater than or equal to • “no more than” – means less than or equal to • “more than” – means greater than • “less than” – means less than

47. Money Applications • Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets. • Write an inequality that represents Keith’s situation. • How many weeks can Keith withdraw money from his account?

48. Money Applications • Yellow Cab Taxi charges a $1.75 flat rate in addition to $0.65 per mile. Katie has no more than $10 to spend on a ride. • Write an inequality that represents Katie’s situation. • How many miles can Katie travel without exceeding her limit?

49. Money Applications • Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend. • Write an inequality that represents Chris’ situation. • How many DVDs can Chris order without exceeding his $100 limit?

50. Money Applications • Skate Land charges a $50 flat fee for birthday party rental and $5.50 for each person. Joann has no more than $100 to spend on the birthday party. • Write an inequality that represents Joann’s situation. • How many people can Joann invite to her birthday party without exceeding her limit?