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{… , – 3, – 2, – 1, 0, 1, 2, 3, …}

Exercise. Use ellipses and set notation to list the set of all integers. {… , – 3, – 2, – 1, 0, 1, 2, 3, …}. Exercise. Use algebraic expressions to represent three consecutive integers. x , x + 1, x + 2. Exercise. Use algebraic expressions to represent three consecutive odd integers.

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{… , – 3, – 2, – 1, 0, 1, 2, 3, …}

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  1. Exercise Use ellipses and set notation to list the set of all integers. {… , – 3, – 2, – 1, 0, 1, 2, 3, …}

  2. Exercise Use algebraic expressions to represent three consecutive integers. x, x + 1, x + 2

  3. Exercise Use algebraic expressions to represent three consecutive odd integers. x, x + 2, x + 4

  4. Exercise Ann has four fewer lamps than Emily, and together they have more than 20 lamps. What is the minimum number of lamps Emily could own? I = the number of Emily’s lamps; (I – 4) + I > 20

  5. Exercise Larry wants to spend no more than $6 for lunch. If he has already purchased a drink and tater tots for $2.45, how many hot dogs can he purchase if they cost $1.25? h = the number of hot dogs; 2.45 + 1.25h ≤ 6

  6. 4 4 Example 1 Mr. Tobin wants to buy four new tires and spend less than $250. What is the range of prices he can spend per tire? Let n = cost per tire. 4n < 250 Mr. Tobin must spend less than $62.50 per tire. n< 62.50

  7. Consecutive Integers integers: n, n + 1, n + 2, etc. even integers: n, n + 2, n + 4, etc. odd integers: n, n + 2, n + 4, etc.

  8. Example 2 The sum of three consecutive integers is more than 20. What are the smallest possible values for the integers? Let n = the first integer. Let n + 1 = the second integer. Let n + 2 = the third integer.

  9. 23 n > 5 3 3 n + (n + 1) + (n + 2) > 20 3n + 3 > 20 3n + 3 – 3 > 20 – 3 3n > 17

  10. The smallest possible integer value for n is 6. The three consecutive integers are 6, 7, and 8. Check: 6 + 7 + 8 > 20 21 > 20

  11. Example 3 Four times the smaller of two consecutive odd integers is less than three times the larger. What are the largest possible values for the integers? Let n = the first odd integer. Let n + 2 = the second odd integer.

  12. 4n < 3(n + 2) 4n < 3n + 6 4n– 3n< 3n– 3n + 6 n < 6 The largest odd integer for n is 5. The second integer would be 5 + 2 = 7. Check: 4(5) < 3(7) 20 < 21

  13. Trichotomy Axiom For all real numbers a and b, a > b, a < b, or a = b.

  14. “less than, equal to, greater than” < = >

  15. ≥ <

  16. ≤ >

  17. –5 –5 Example 4 Solve –5x + 7 ≤ 22. –5x + 7 > 22 –5x + 7 – 7 > 22 – 7 –5x> 15 x < –3

  18. > “is at most” or “is not more than” or “is the maximum amount” Common Wording Negated is less than or equal to Meaning ≤ Equivalent

  19. < “is at least” or “is not less than” or “is the minimum amount” Common Wording Negated is greater than or equal to Meaning ≥ Equivalent

  20. Example 5 Write the corresponding inequality for this statement: four more than a number is at most 84. n + 4 ≤ 84

  21. Example 5 Write the corresponding inequality for this statement: twice a number decreased by 7 is at least 90. 2x – 7 ≥ 90

  22. Example 5 Write the corresponding inequality for this statement: forty less than a number is not less than 95. y – 40 ≥ 95

  23. Example 5 Write the corresponding inequality for this statement: six more than a number is not equal to 63. x + 6 ≠ 63

  24. Example 6 Noah and Kim are planning to attend the state fair. They will pay $5 to park the car and $2.50 per event ticket. What is the range of the number of event tickets that they can purchase if they cannot spend more than $36 at the fair?

  25. 2.5 2.5 Let n = the number of tickets. 5 + 2.5n ≤ 36 5 – 5 + 2.5n ≤ 36 – 5 2.5n ≤ 31 n ≤ 12.4 Since the number of tickets must be a whole number, they can purchase up to 12 tickets.

  26. Exercise If Angelica can get 10% interest compounded annually, how much must she put aside to have at least $1,000 in savings a year from now? P(1 + 0.1) ≥ 1,000 P ≥ $909.09

  27. Exercise If she puts aside the same amount, P, each year, how much must she put aside each year to have at least $1,000 in savings two years from now? P(1.1) + P(1.1)2 ≥ 1,000 P ≥ $432.90

  28. Exercise If peanuts cost $1/lb. and cashews cost $2.25/lb., how much of each should go in a 50 lb. mixture if you want to sell it for no more than $1.75/lb.? n = pounds of peanuts 1.00n + 2.25(50 – n) ≤ 1.75(50) 20 lb. peanuts; 30 lb. cashews

  29. Exercise If the bottom of the B range is 80% and Greg has scores of 73 and 85, what must he average on the last two tests to get a B or better in the class? x = the average 73 + 85 + 2x ≥ 4(80) x ≥ 81

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