Sect. 1.1 Some Basics of Algebra Numbers, Variables, and Constants Operations and Exponents

English phrases for operations Algebraic Expressions vs. Equations Evaluating Algebraic Expressions Sets and Set Notation Important Sets of Numbers

11.1

Numbers, Variables, and Constants Numbers: 127, 4.39, 0, -11¾, square root of 3

Integers, Decimals, Fractions, Mixed Numbers

Variables: x, a, b, y, Q, B2 etc

Constants: π, e, C=speed of light in vacuum

21.1

Operations and Exponents Operations combine two numbers

Addition 3 + 6.2 Subtraction ⅔ – 5 Multiplication 356 · 0.03 or 356(0.03) Division 19 / 3 or 19 ÷ 3

Exponents 74 Short for 7·7·7·7

31.1

Class Exercise: Op’s + – • 6 + 4 + 3 + 7 + 9 + 1 = 30 9 + 2 + 1 + 3 + 8 = 23 (-6) + (-2) + (-5) = -13 -6 – 2 – 5 = -13 8 + (-2) + (-9) + 6 + (-4) = 14 + (-15) = -1 6 • 2 • 5 = 60 -3 • 7 • (-2) = 42 2 • (-5) • (-3) • (-4) = -120

41.1

13

7

13

3

13

4

30

1

30

5202430

56

51

103

102

65

64

6

1

3

2

5

4

lcd

4

1

6

5

10

3

Class Exercise: Op ÷, fractions

20

3

5

2

8

3

2

5

8

3

51.1

Algebraic Expressions vs. Equations

Algebraic expressions have one or more terms Sometimes expressions can be simplified If each variable is replaced with a number, we can evaluate an

expression (reduce it to a single number) Today we will review how to evaluate expressions Tomorrow we’ll look at equations

An equation is two expressions separated by an equal sign – equations are not evaluated, they are solved 61.1

Evaluating Algebraic Expressions Substitution is replacing a variable with a number When every variable in an expression is substituted

with a number, we can evaluate that expression Evaluate 3xz + y for x = 2, y = 5, and z = 7

3xz + y (write original problem) 3(2)(7) + (5) (put parentheses for each variable) (insert the corresponding numbers) 42 + 5 (simplify according to “order of operations”) 47 (final answer)

71.1

Class Exercise: mixed + • – ÷ 3 + 2 • 6 = ?

5 • 6 = 30 or 3 + 12 = 15

-3 – 3 = ? -6 or 0

3 • 22 = ? 62 = 36 or 3 • 4 = 12

6 + 4 ÷ 2 = ? 10 ÷ 2 = 5 6 + 2 = 8

81.1

Rules for Order of Operations To make sure an expression is always evaluated in

the same way by different people, the Order of Operations convention was defined

Mnemonic: “Please Excuse My Dear Aunt Sally”

Parentheses Exponents Multiply/Divide

Add/Subtract

Always: Evaluate & Eliminate the innermost grouping first91.1

Order of Ops Example 2 { 9 – 3 [ -2x – 4 ] } 2 { 9 + 6x + 12 } 2 { 6x + 21} 12x + 42 Remember: It’s an INSIDE job

101.1

Class Exercise – Evaluate expressions 7x + 3 for x = 5

7(5) + 3 35 + 3 38

3z – 2y for y = 1 and z = 6 3(6) – 2(1) 18 – 2 16

[17 – (a – b)] for a = -3 and b = 7 [17 – (-3 – 7)] [17 – (-10)] 17 + 10 27 111.1

Sets and Set Notation Finite sets and Infinite sets Roster notation: {1, 2, 3, … } with ellipsis

Set-Builder notation: { x | x is an integer > 0} Set of all real numbers: Empty Set (no members): Element of a set: 5 {1, 2, 3, 4, 5, 6} Union of sets: {1, 2, 3} {3, 4, 5} = {1, 2, 3, 4, 5} Intersection of sets: {1, 2, 3} {3, 4, 5} = { 3 } Subset of a set: {1, 2, 3} {1, 2, 3, 4, 5}

121.1

Different Sets of Numbers

131.1

Next time: 1.2 Operations and Properties

of Real Numbers

141.1