2. Confidential2 1) 81 > b -90 b < 171 2) s -12 > 8 s > 20 3) n - 3 ≤ 8 n ≤ 11 4) j – 7 ≥ 5 j ≥ 12 5) f - 83 < - 56 f < 27 Warm up

3. Confidential3 An inequality is a statement with a symbol of or ≥ between numerical or variable expressions. Example: x + 7 ≤ 11. x is a variable. x x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d,....) to be a variable Defining Inequality Lets review what we have learnt in the last lesson

4. Confidential4 Inequality Symbols SYMBOL MEANING < less than > greater than ≤ less than or equal to ≥ greater than or equal to

5. Confidential5 Rule No : 1 - Addition If a < b, then a + c < b + c Rules to Solve Linear Inequalities Rule No : 2 - Subtraction If a < b, then a - c < b - c

6. Confidential6 ADDING & SUBTRACTING INEQUALITY Steps:- To solve the inequality means isolate a variable x and move the numbers to the other side of the inequality. To move the number perform the reverse operation. Addition is the reverse of Subtraction Subtraction is the reverse of Addition

7. Confidential7 Word problems in Inequalities PhrasesTranslation is at least ≥ is at most ≤ cannot exceed ≤ must exceed > is between < and < To solve the word problem translate the phrases and solve it

8. Confidential8 Just like with equations, the solution to an inequality is a value that makes the inequality true. You can solve inequalities in the same way you can solve equations, by following these rules.

9. Confidential9 If c is positive and a < b, then ac < bc and a b, c = 0 c c 3. If c is positive and a > b, then ac > bc and a b, c = 0 c c < > For all numbers a, b and c, the following rules are true

10. Confidential10 a (x) a (5) ? If x < 5 is a (x) < a (5) ? Multiplication

11. Confidential11 (x) a (5) a ? If x < 5 is (x) < (5) a a Division

12. Confidential12 Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by 16 in order to leave x alone on the left side. 16x > 512 16 x > 512 16 x > 32

13. Confidential13 Examples 8x < 296 8x ÷ 8 < 296 ÷ 8 x < 37 y 4 < 45 y 1 4 < 45 ÷ ÷ Y < 180 1414 Use Division Use Multiplication

14. Confidential14 When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol.

15. Confidential15 If c is negative and a < b, then ac > bc and a b c c If c is negative and a > b, then ac < bc and a b c c c = 0 > < Opposite sign

16. Confidential16 x 5 If x < 5, what about the opposite of x and 5? -x -5 Opposite sign

17. Confidential17 x 5 If x < 5, what about the opposite of x and 5? The answer: -5 -x - x > -5 It’s the opposite!

18. Confidential18 So if you multiply or divide by a negative number: REVERSE the s ss symbol !

19. Confidential19 If there is a negative number on one side of the inequality then we need to multiply/ divide by a negative number and reverse the symbol Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by –16 in order to leave x alone on the left side. 16/16 x > 512/16 When we simplify, because we're dividing by a negative number, we have to remember to reverse the symbol. This gives "x is less than –32," not "x is greater than –32.“ x < -32 -16 x > 512

20. Confidential20 Example 45 < 60 Perform the multiplication operation on each side of the equality Multiply by 2 45 < 60 45(2) < 60(2) 90 < 120 True Multiply by -2 45 < 60 45(-2) < 60(-2) -90 < -120 False -90 > -120 True

21. Confidential21 Example 45 < 60 Perform the division operation on each side of the equality Divide by 5 Divide by -5 45 < 60 45 60 5 5 < 9 < 12 45 < 60 45 60 -5 < -9 < -12 True False -9 > -12True

22. Confidential22 Your Turn ! Solve each Inequality 1. 5c < 20 c < 4 2. 6m > 12 m > 2 3. 9j > 81 j > 9 4. Three times a number is more than 18 n > 6 5. $8 was less than 2/3 of the total m < 12 money spent

23. Confidential23 Your turn 6.-11b ≥ -30 b ≤ 30/11 7.y/-5 > 12 y < -60 8. -868 < -16r r < 217/4 9.f/17 ≥ 20.2 f ≥ 343.4 10. 34 > 9z z < 34/9

24. Confidential24

25. Confidential25 Game time Click here to play a gameplay

26. Confidential26 1. Alyssa's boss told her that if her department's income decreases more than eleven percent from last year's income she will not receive a bonus. Alyssa's department made $137,295 for the company last year. At the end of this year, the company accountant told Alyssa that her department's income was $124,938. Will Alyssa receive a bonus this year? yes, she will receive a bonus as her department ‘s income decreased 9 %

27. Confidential27 2. Jeb wants to rent a car for his vacation. Value Cars rents cars for $25 per day plus $0.25 per mile. How far can he drive for one day if he wants to spend no more that $200 on car rental? up to 700 miles

28. Confidential28 3. Alida has at most $10.50 to spend at a convenience store. She buys a bag of potato chips and a can of soda for $1.55. If gasoline at this store costs $1.35 per gallon, how many gallons of gasoline can Alida buy for her car, to the nearest tenth of a gallon? 1.55 + 1.35g ≤ 10.50 g ≤ 6.6 Alida will have enough money if she gets no more than 6.6 gallons of gasoline.

29. Confidential29 Just like with equations, the solution to an inequality is a value that makes the inequality true. You can solve inequalities in the same way you can solve equations, by following these rules.

30. Confidential30 For all numbers a, b and c, the following rules are true 1.If c is positive and a < b, then ac < bc and a b, c = 0 c c 3. If c is positive and a > b, then ac > bc and a b, c = 0 c c < >

31. Confidential31 Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by 6 in order to leave x alone on the left side. 18 > 6k 18 6 > 6k 6 k < 3

32. Confidential32 When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol. When we simplify, because we're dividing by a negative number, h < -3 -4h > 12 4 4

33. Confidential33 You had a great lesson today ! Be sure to practice what you have learned