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. I = the number of Emily’s lamps; (I – 4) + I > 20
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. h = the number of hot dogs; 2.45 + 1.25h ≤ 6
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; h ≤ 6

6. Mr. Tobin must spend less than $62.50 per tire.
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. 4
n < 62.50

7. even integers: n, n + 2, n + 4, etc.
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. n + (n + 1) + (n + 2) > 20
3n + 3 > 20
3n + 3 – 3 > 20 – 3
3n > 17 3
n > 5
2 3

10. The smallest possible integer value for n is 6
The smallest possible integer value for n is 6. The three consecutive integers are 6, 7, and 8.
Check: > 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 = 7.
Check: 4(5) < 3(7)
20 < 21

13. For all real numbers a and b, a > b, a < b, or a = b.
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. Example 4 Solve – 5x + 7 ≤ 22. – 5x + 7 > 22

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. Let n = the number of tickets.
2.5
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( ) ≥ 1,000
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 (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
x ≥ 4(80)
x ≥ 81