1. Solving and Graphing Inequalities CHAPTER 6 REVIEW

2. An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

3. “x < 5” means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!

4. Numbers less than 5 are to the left of 5 on the number line. 051015-20-15-10-5-252025 If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. The number 5 would not be a correct answer, though, because 5 is not less than 5.

5. “x ≥ -2” means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to - 2!

6. Numbers greater than -2 are to the right of 5 on the number line. 051015-20-15-10-5-252025 If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”. -2

7. Where is -1.5 on the number line? Is it greater or less than -2? 051015-20-15-10-5-252025 -1.5 is between -1 and -2. -1 is to the right of -2. So -1.5 is also to the right of -2. -2

8. Solve an Inequality w + 5 < 8 w + 5 + (-5) < 8 + (-5) w < 3 All numbers less than 3 are solutions to this problem!

9. More Examples 8 + r ≥ -2 8 + r + (-8) ≥ -2 + (-8) r ≥ -10 All numbers from -10 and up (including -10) make this problem true!

10. More Examples x - 2 > -2 x + (-2) + (2) > -2 + (2) x > 0 All numbers greater than 0 make this problem true!

11. More Examples 4 + y ≤ 1 4 + y + (-4) ≤ 1 + (-4) y ≤ -3 All numbers from -3 down (including -3) make this problem true!

12. There is one special case. ● Sometimes you may have to reverse the direction of the inequality sign!! ● That only happens when you multiply or divide both sides of the inequality by a negative number.

13. Example: Solve: -3y + 5 >23 -5 -5 -3y > 18 -3 -3 y < -6 ● Subtract 5 from each side. ● Divide each side by negative 3. ● Reverse the inequality sign. ● Graph the solution. 0-6

14. Try these: 1.) Solve 2x + 3 > x + 5 2.)Solve - c – 11 >23 3.) Solve 3(r - 2) < 2r + 4 -x x + 3 > 5 -3 -3 x > 2 + 11 -c > 34 -1 -1 c < -34 3r – 6 < 2r + 4 -2r r – 6 < 4 +6 +6 r < 10

15. You did remember to reverse the signs...... didn’t you?Good job!

16. Example: - 4x + 6 +6 -2 -2 Ring the alarm! We divided by a negative! We turned the sign!

17. Remember Absolute Value

18. Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

19. Ex: Solve 2x + 7 -3 = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

20. Ex: Solve & graph. Becomes an “and” problem -3 7 8

21. Solve & graph. Get absolute value by itself first. Becomes an “or” problem -2 3 4

22. Example 1: ● |2x + 1| > 7 ● 2x + 1 > 7 or 2x + 1 >7 ● 2x + 1 >7 or 2x + 1 <-7 ● x > 3 or x < -4 This is an ‘or’ statement. (Greator). Rewrite. In the 2 nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. 3 -4

23. Example 2: ● |x -5|< 3 ● x -5< 3 and x -5< 3 ● x -5 -3 ● x 2 ● 2 < x < 8 This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. 8 2

24. Absolute Value Inequalities Case 1 Example: and

25. Absolute Value Inequalities Case 2 Example: OR or

26. Absolute Value Answer is always positive Therefore the following examples cannot happen... Solutions: No solution

33. Sean earns $6.70 per hour working after school. He needs at least $155 (≥ 155) for a stereo system. Write an inequality that describes how many hours he must work to reach his goal. $6.70 h ≥ 155 Per means to multiply To figure your pay check: You multiply rate times hours worked So 23 hours is not enough So Sean must Work 24 hours To earn enough Money.

34. Sean earns $6.85 per hour working after school. He needs at least $310 (≥ 310) for a stereo system. Write an inequality that describes how many hours he must work to reach his goal. $6.85 h ≥ 310 Per means to multiply To figure your pay check: You multiply rate times hours worked So 45 hours is not enough So Sean must Work 46 hours To earn enough Money.

35. The width of a rectangle is 11 cm. Find all possible values for the length of the rectangle if the perimeter is at least 300 cm. ( ≥ 300 cm) 2L + 2W = p

36. The width of a rectangle is 31 cm. Find all possible values for the length of the rectangle if the perimeter is at least 696 cm. ( ≥ 696 cm) 2L + 2W = p

37. The perimeter of a square is to be between 11 and 76 feet, inclusively. Find all possible values for the length of its sides. 4 equal sides Perimeter – add up sides s s s s P = s +s+s+s=4s

38. The perimeter of a square is to be between 14 and 72 feet, inclusively. Find all possible values for the length of its sides. 4 equal sides Perimeter – add up sides s s s s P = s +s+s+s=4s

39. Three times the difference of a number and 12 is at most 87. Let x represent the number and find all possible values for the number.

40. Marlee rented a paddleboat at the Park for a fixed charge of $2.50 plus $1.50 per hour. She wants to stay out on the water as long as possible. How many hours can she use the boat without spending more than $7.00?