1. Exercise
2x − 3 = 9
x = 6

2. Exercise
−2x + 3 = 3
x = 0

3. Exercise
2x + 3 = 4
x = 0.5

4. Exercise
= −6
x − 3 2
x = −9

5. Exercise
− 3 = 6
x 2
x = 18

7. Natural Numbers Natural numbers are counting numbers.
= {1, 2, 3, 4, 5…}

8. Whole Numbers Whole numbers are natural numbers and zero.
= {0, 1, 2, 3, 4, 5…}

9. N is a subset of W.

10. 100 2 17

11. Integers Integers are whole numbers and opposites of naturals.
= {...−3, −2, −1, 0, 1, 2, 3…}

12. N and W are subsets of Z.

13. 100 2 17 −4
−26 −8

14. Rational Numbers
Rational numbers are integers and all fractions.

15. = {
a b a
b & b ≠ 0 } ,

17. 100 2 17 −4
−26 −8
2.58
−0.7
1 7
2 7 −

18. Irrational Numbers
Irrational numbers are totally different from rational numbers. The two have nothing in common.

19. Rationals and irrationals are disjoint sets.
In other words, they have no common element.

20. Irrationals
2, , 5 7, p

21. Real Numbers
Real numbers include both rational and irrational.

22. = & irrationals

23. 100 2 17 −4
−26 −8
2.58
−0.7
1 7
2 7 −
Irrationals 12

24. Example 1
Place the given numbers in the correct set on a Venn diagram. Be as specific as possible when classifying
them:
−15; 0; ; 1,290; − ;
12.
3 2
2 5

25. −15; 0; ; 1,290; − ; 12
3 2
2 5
3 2
Irrationals 12
1,290
−15
2 5 −

26. Example 2
Does the following equation have a solution over the set given? If yes, solve.
3x = 39 over the set of whole numbers
Yes; dividing both sides by 13 produces x = 13, a whole number.

27. Example 2
Does the following equation have a solution over the set given? If yes, solve.
7y = 48 over the set of integers
No; dividing both sides by 7 produces a rational number that is not an integer.

28. A “>” means “greater than” something else.

29. A “<” means “less than” something else.

30. A “≥” means “greater than or equal to” something else.

31. A “≤” means “less than or equal to” something else.

32. A “≠” means “not equal to” something else.

33. Inequalities may have many solutions or no solution.
All equations where the variable has an exponent of 1 have only one solution.
Inequalities may have many solutions or no solution.

34. Inequality
An inequality is a mathematical sentence expressing the relative size of two quantities that may or may not be not equal.

35. All the numbers that make an inequality true form the solution set.

36. Example 3
Which numbers are members of the solution set for the inequality 3x < −6: −5, −3, −1, 0, or 2?
3(−5) = −15 < −6; true
3(−3) = −9 < −6; true
3(−1) = −3 < −6; false
3(0) = 0 < −6; false
3(2) = 6 < −6; false
3(−5) = −15 < −6; true
3(−3) = −9 < −6; true
3(−1) = −3 < −6; false
3(0) = 0 < −6; false
3(2) = 6 < −6; false

37. Example
Does the following equation have a solution over the set given? If yes, solve.
−4x = 20; whole numbers

38. Example
Does the following equation have a solution over the set given? If yes, solve.
−3 + x = 10; whole numbers

39. Example
Does the following equation have a solution over the set given? If yes, solve.
= 4; integers
x 3

40. Example
Does the following equation have a solution over the set given? If yes, solve.
5m = 8; integers

41. Example
Does the following equation have a solution over the set given? If yes, solve.
x2 = 10; rational numbers

42. Example
Does the following equation have a solution over the set given? If yes, solve.
4x = 15; rational numbers