1. CONFIDENTIAL 1 Solving Inequalities by Multiplying or Dividing Solving Inequalities by Multiplying or Dividing

2. CONFIDENTIAL 2 Warm Up Solve each inequality and graph the solutions. 1) n - 15 < 3 2) m - 13 > 29 3) v – 4 < 7 4) t – 5 > 11

3. CONFIDENTIAL 3 Inequalities Solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number. STEP1: Identify the variable. STEP2: To get the variable by itself, Multiply the same number to or Divide the same number from each side of the inequality. STEP3: Check the solution. Solving inequality by Multiply or Divide needs certain steps to be followed.

4. CONFIDENTIAL 4 The rules are similar for a ≥ b and a ≤ b. Multiplication property of inequality When you multiply each side of a true inequality by a positive integer, the result remains true. In symbol: For all integers a, b, and c, where c > 0. 1. If a > b, then a × c > b × c and 2. If 7 > 2, then 7 × 4 > 2 × 4 Multiplication and Division by Positive Numbers

5. CONFIDENTIAL 5 Division property of inequality The rules are similar for a ≥ b and a ≤ b. When you divide each side of a true inequality by a positive integer, the result remains true. In symbol: For all integers a, b, and c, where c > 0. 1. If a > b, then a > b, c c 2. If 7 > 3, then 7 > 3, 4 4

6. CONFIDENTIAL 6 Multiplying or Dividing by a Positive Number Solve each inequality and graph the solutions. A) 3x > -27 3x > -27 3 3 1x > -9 x > -9 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x > -9

7. CONFIDENTIAL 7 B) 2r < 6 3 r < 9 Since r is multiplied by 2/3, multiply both sides by the reciprocal of 2/3. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 r < 9 2r < (6) 3 3232 3232 2r < 6 3

8. CONFIDENTIAL 8 Solve each inequality and graph the solutions. Now you try! 1) 4k > 24 2) g > 27 3434

9. CONFIDENTIAL 9 If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true. 5 > -3 5 is greater than -3. Multiply both sides by -2. You know that -10 is less than 6, so use the symbol for less than. 5 (-2) -3 (-2) -10 6 -5 < 3

10. CONFIDENTIAL 10 Multiplying both sides by a negative number changes the sign of both sides of the inequality. This means there is another set of properties of inequality for multiplying or dividing by a negative number. 0 2 4 6 8 10 12 10 86 4 2 5 1214 3 x(-2)

11. CONFIDENTIAL 11 The rules are similar for a > b and a < b. When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol. In symbol: For all integers a, b, and c, where c < 0. 1. If a > b, then a x c < b x c and 2. If 7 > 3, then 7 x -5 < 3 x -5 Multiplication property of inequality Multiplication and Division by Negative Numbers

12. CONFIDENTIAL 12 The rules are similar for a > b and a < b. When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol. In symbol: For all integers a, b, and c, where c < 0. 1. If a > b, then a < b and c c 2. If 7 > 2, then 7 < 2, -4 -4 Division property of inequality Multiplication and Division by Negative Numbers

13. CONFIDENTIAL 13 A) -8x > 72 -8x > 72 x < -9 Since x is multiplied by -8, divide both sides by -8. Change > to <. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < -9 Multiplying or Dividing by a Negative Number Solve each inequality and graph the solutions. -8x < 72 -8 -8

14. CONFIDENTIAL 14 B) -3 ≤ x -5 (-5) -3 ≥ (-5)x -5 15 ≥ x (or x ≤ 15) Since x is divided by -5, multiply both sides by -5. Change ≤ to ≥. 01 2 34 5 678 9 1011 12 13 14 1516 1718 19 -3-2 x ≤ 15 20 21

15. CONFIDENTIAL 15 Solve each inequality and graph the solutions. Now you try! 1) 10 ≥ -2x 2) 4.25 > -0.25h

16. CONFIDENTIAL 16 Problem Solving Application Ryan has a $16 gift card for a health store where a smoothie costs $2.50 with tax. What are the possible numbers of smoothies that Ryan can buy? Let s represent the number of smoothies Ryan can buy. $2.50 times number of smoothies is at most $16.00. 2.50 s ≤ 16.00 s ≤ 6.4 2.50 s ≤ 16.00 2.50 Since s is multiplied by 2.50, divide both sides by 2.50. The symbol does not change. Ryan can buy only a whole number of smoothies. Ryan can buy 0, 1, 2, 3, 4, 5, or 6 smoothies.

17. CONFIDENTIAL 17 1) A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher can fill? Now you try!

18. CONFIDENTIAL 18 Assessment Solve each inequality and graph the solutions. 1) 10 < 2t 3) -80 < 8c 4) 21 > 3d 2) j ≤ 4 1313 6) h ≤ 2 4 7 5) ≥ -2 w4w4

19. CONFIDENTIAL 19 Write an inequality for each sentence. Graph each inequality. 7) The product of a number and 7 is not less than 21. 8) The quotient of h and -6 is at least 5.

20. CONFIDENTIAL 20 9) The rope Rosa brought with her camping gear is 54 inches long. Rosa needs to cut shorter pieces of rope that are each 18 inches long. What are the possible number of pieces Rosa can cut? 10) What is the greatest possible integer solution of the inequality 3.806x < 19.902?

21. CONFIDENTIAL 21 Inequalities Solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number. STEP1: Identify the variable. STEP2: To get the variable by itself, Multiply the same number to or Divide the same number from each side of the inequality. STEP3: Check the solution. Solving inequality by Multiply or Divide needs certain steps to be followed. Let’s review

22. CONFIDENTIAL 22 The rules are similar for a ≥ b and a ≤ b. Multiplication property of inequality When you multiply each side of a true inequality by a positive integer, the result remains true. In symbol: For all integers a, b, and c, where c > 0. 1. If a > b, then a × c > b × c and 2. If 7 > 2, then 7 × 4 > 2 × 4 Multiplication and Division by Positive Numbers

23. CONFIDENTIAL 23 Division property of inequality The rules are similar for a ≥ b and a ≤ b. When you divide each side of a true inequality by a positive integer, the result remains true. In symbol: For all integers a, b, and c, where c > 0. 1. If a > b, then a > b, c c 2. If 7 > 3, then 7 > 3, 4 4

24. CONFIDENTIAL 24 If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true. 5 > -3 5 is greater than -3. Multiply both sides by -2. You know that -10 is less than 6, so use the symbol for less than. 5 (-2) -3 (-2) -10 6 -5 < 3

25. CONFIDENTIAL 25 Multiplying both sides by a negative number changes the sign of both sides of the inequality. This means there is another set of properties of inequality for multiplying or dividing by a negative number. 0 2 4 6 8 10 12 10 86 4 2 5 1214 3 x(-2)

26. CONFIDENTIAL 26 The rules are similar for a > b and a < b. When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol. In symbol: For all integers a, b, and c, where c < 0. 1. If a > b, then a x c < b x c and 2. If 7 > 3, then 7 x -5 < 3 x -5 Multiplication property of inequality Multiplication and Division by Negative Numbers

27. CONFIDENTIAL 27 The rules are similar for a > b and a < b. When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol. In symbol: For all integers a, b, and c, where c < 0. 1. If a > b, then a < b and c c 2. If 7 > 2, then 7 < 2, -4 -4 Division property of inequality Multiplication and Division by Negative Numbers

28. CONFIDENTIAL 28 A) -8x > 72 -8x > 72 x < -9 Since x is multiplied by -8, divide both sides by -8. Change > to <. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < -9 Multiplying or Dividing by a Negative Number Solve each inequality and graph the solutions. -8x < 72 -8 -8

29. CONFIDENTIAL 29 Problem Solving Application Ryan has a $16 gift card for a health store where a smoothie costs $2.50 with tax. What are the possible numbers of smoothies that Ryan can buy? Let s represent the number of smoothies Ryan can buy. $2.50 times number of smoothies is at most $16.00. 2.50 s ≤ 16.00 s ≤ 6.4 2.50 s ≤ 16.00 2.50 Since s is multiplied by 2.50, divide both sides by 2.50. The symbol does not change. Ryan can buy only a whole number of smoothies. Ryan can buy 0, 1, 2, 3, 4, 5, or 6 smoothies.