3.6 Compound inequalities:

1. Inverse Operations: Operations that undo another operation. 3.6 Compound inequalities: Isolate: The use of inverse operations used to leave a variable by itself. Compound Inequalities: two distinct inequalities joined by the word and or the word or.

2. GOAL:

3. Compound inequalities: These inequalities have the following graphs when the word and is being used: Set Notation: {x|3 ≤ x ≤ 7} Interval: [3, 7]

4. Compound inequalities: These inequalities have the following graphs when the word or is being used: Set Notation: {x| x < -2 or x ≥1} Interval: (-∞,-2)U[1,∞)

5. WRITING Compound inequalities: We must be able to transform English sentences into math equations: EX: What compound inequality represents the phrase: “ All real numbers that are greater than – 2 and less than or equal to 6”

6. “ All real numbers that are greater than – 2 and less than or equal to 6” SOLUTION: All real numbers greater than -2 X > -2  and -2 < x ≤ 6 All real numbers less than or equal to 6  X ≤ 6 Set Notation: {x| -2 < x ≤ 6} Interval: (-2, 6]

7. YOU TRY IT: What compound inequality represents the phrase: “ All real numbers that are less than 0 or greater than 5”

8. “ All real numbers that are less than 0 or greater than 5” SOLUTION: All real numbers less than 0 X < 0  x< 0, x > 5 or All real numbers greater than 5  X > 5 Set Notation: {x| x < 0, x > 5} Interval: (-∞,0) U (5, ∞)

9. SOLVING Compound Inequalities: To solve inequalities we follow inverse operations just like we do to isolate a variable: EX: What are the solutions of:- 2 ≤ 2m – 4 < -1

10. SOLUTION: - 2 ≤ 2m – 4 < -1 Given (and) 4 - 2 ≤ 2m <-1 + 4 Inverse of subt. 2 ≤ 2m < 3 Like terms Inverse of mult. ≤ m < 1 ≤ m < 1.5 Interval: [1, 1.5)

11. YOU TRY IT: What are the solutions to 3t + 2 < -7 or -4t + 5 < 1

12. SOLUTION: or -4t + 5 < 1 3t + 2 < -7 3t < -7 -2 -4t < 1 - 5 -4t < -4 3t < -9 t < -3 t > 1 Interval: (-∞,-3) U (1,∞)

13. Real-World: You have taken a quiz and got 55%. You are about to take another quiz next week. If you want to pass the portion of quizzes in the class you must get an average between 70% and 79% What are the possible percentages you must get on the next quiz?

14. Real-World: (SOLUTION) Quiz 1 = 55% Quiz 2 = x% 70% Average  140 70% and 79% 85 Thus in order for you to obtain a passing grade in your quizzes, you must get between 85% and 103% on your next quiz? (Good Luck!!!)

15. Real-World: A secondary 15-year old student should consume no more than 2200 calories per day. A moderately active student should consume between 2400 and 2800. An active 15-year-old student should consume between 2800 and 3200 calories per day. Model these ranges on a number line and represent them in set and interval notations.

16. SOLUTION: A secondary 15-year old student should consume no more than 2200 calories per day. 2200 1000 2000 0 Set: { x | 0 < x ≤ 2200} Interval: (0, 2200]

17. SOLUTION: A moderately active student should consume between 2400 and 2800 Calories per day. 2400 2800 0 1000 2000 3000 Set: { x | 2400 < x ≤ 2800} Interval: (2400, 2800)

18. SOLUTION: An active 15-year-old student should consume between 2800 and 3200 calories per day. 3200 2800 0 1000 2000 3000 Set: { x | 2800 < x ≤ 3200} Interval: (2800, 3200)

19. CLASSWORK:Page 204-205 Problems: 1, 2, 5, 6, 11, 14, 16, 20, 23, 25, 43.