1. Algebra 1: Topic 1 Notes

2. Table of Contents Order of Operations & Evaluating Expressions
The Real Number System
Properties of Real Numbers
Simplifying Radicals
Basic Exponent Properties
One & Two-Step Equations
Basic Multi-Step Equations

3. Order of Operations & Evaluating Expressions

4. Review: Order of Operations
Please- Parentheses Excuse- Exponents My- Multiplication Dear- Division Aunt- Addition Sally- Subtraction

5. Review: Order of Operations
Remember that Multiplication and Division are a pair…do in order from left to right. Addition and Subtraction are also a pair….do in order from left to right.

6. Review: “Evaluate” Evaluating an expression means to plug in!
For example:
2n + 4 for n = 3
2(3) + 4 10

7. Let’s Practice… Evaluate the following expressions.
1. 3r – 17 for r = – 6d for d = 4
3. 5g + 20 – (4g) for g = (3 + p)2 – 2p for p = 7

8. The Real Number System: Numbers Have Names?!?!

10. Natural Numbers
Non-decimal, positive numbers starting with one.

11. Whole Numbers
Non-decimal, positive numbers and zero.

12. Integers
Non-decimal positive and negative numbers, including zero.

13. Rational Numbers
Any number that can be expressed as the quotient or fraction 𝑝 𝑞 of two integers.
YES:
Any integers
Any decimals that
ends or repeats
Any fraction
NO:
Never ending decimals

14. Irrational Numbers Any number that can not be expressed as a fraction.
Usually a never-ending, non-repeating decimal.
Examples: 𝜋
2 , 5 ….

15. Let’s Practice…
Rational or Irrational.
2 17 22
3 27
1 3

16. Properties of Real Numbers

17. List of Properties of Real Numbers
Commutative
Associative
Distributive
Identity
Inverse

18. Commutative Properties
Definition: Changing the order of the numbers in addition or multiplication will not change the result.

19. Commutative Properties
Property
Numbers
Algebra
Addition
Multiplication

20. Associative Properties
Definition: Changing the grouping of the numbers in addition or multiplication will not change the result.

21. Associative Properties
Property
Numbers
Algebra
Addition
Multiplication

22. Distributive Property
Multiplication distributes over addition.

23. Additive Identity Property
Definition: Zero preserves identities under addition. In other words, adding zero to a number does not change its value. Example:

24. Multiplicative Identity Property
Definition: The number 1 preserves identities under multiplication. In other words, multiplying a number by 1 does not change the value of the number. Example:

25. Additive Inverse Property
Definition: For each real number a there exists a unique real number –a such that their sum is zero. In other words, opposites add to zero. Example:

26. Multiplicative Inverse Property
Definition: For each real number a there exists a unique real number 1 𝑎 such that their product is 1.
Example:

27. Simplifying Radicals

28. Radical Vocab

29. How to Simplify Radicals
Make a factor tree of the radicand.
Circle all final factor pairs.
All circled pairs move outside the radical and become single value.
Multiply all values outside radical.
Multiply all final factors that were not circled. Place product under radical sign.

31. Let’s Practice…

32. Let’s Practice…

33. How to Simplify Cubed Radicals
Make a factor tree of the radicand.
Circle all final factor groups of three.
All circled groups of three move outside the radical and become single value.
Multiply all values outside radical.
Multiply all final factors that were not circled. Place product under radical sign.

34. Let’s Practice…

35. Let’s Practice…

36. Exponents

37. Definition: Exponent
The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to the right and above the base number.

38. The Zero Exponent Rule
Any number (excluding zero) to the zero power is always equal to one.
Examples:
1000=1
1470=1
550 =1

39. Negative Power Rule

40. Let’s Practice…
5-2
3 4 −2
(-3)-3

41. The One Exponent Rule
Any number (excluding zero) to the first power is always equal to that number.
Examples:
a1 = a
71 = 7
531 = 53

42. Simplifying Exponential Expressions
Step 1: Simplify (get rid of negatives and fractions, where possible). Step 2: Plug in.

43. Let’s Practice… 4m3j-2 2. 8b3 4a −2
for m = -2 and j = 3 for b = 2 and a = 3

44. Solving 2-step Equations

46. Inverse Operations Pairs
Addition
Subtraction
Multiplication
Division
Exponents
Radicals

47. How to Solve 2-Step Equations
Use the opposite operation to move the letters to the left and numbers to the right.
Divide to get the variable alone.

48. Example
5x - 18 = 22

49. Let’s Practice…
10x – 7 = – 3m = 45

50. Let’s Practice…
3. 4y = 3y b = 17 – 8b

51. Fractions…
To get rid of fractions, multiply all other values by the reciprocal! Example:

52. Let’s Practice…
− 2 5 x = 6

53. Solving Multi-Step Equations

54. How to solve Multi-Step Equations
Distribute.
Bonus Step: Multiply by reciprocal/LCM, if fractions.
Combine Like Terms (ONLY on same side of equal sign).
Use the inverse operation to move numbers to the right.
Use the inverse operation to move variables to the left.
Divide.

55. Example
3(4x – 10) = 26

56. Example
6(4 + 3x) - 10x = x

57. Example
1 5 x – 8 = 4 + 2x

58. Let’s Practice…
4(5 + 2x) + 7x = -5

59. Let’s Practice…
3 2 𝑥+6=− 𝑥