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Sect. 1.1 Some Basics of Algebra

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. Numbers, Variables, and Constants.

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Sect. 1.1 Some Basics of Algebra

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  1. 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 1.1

  2. 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 1.1

  3. 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 1.1

  4. 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 1.1

  5. Class Exercise: Op ÷, fractions 1.1

  6. 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 1.1

  7. 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) 1.1

  8. 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 1.1

  9. 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 first 1.1

  10. Order of Ops Example • 2 { 9 – 3 [ -2x – 4 ] } • 2 { 9 + 6x + 12 } • 2 { 6x + 21} • 12x + 42 • Remember: It’s an INSIDE job 1.1

  11. 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 1.1

  12. 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} 1.1

  13. Different Sets of Numbers 1.1

  14. Next time: • 1.2 Operations and Propertiesof Real Numbers 1.1

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