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Review Chapter

This chapter covers the basic components of algebra, including expressions, inequalities, equations, sets, and the classification of real numbers. Learn the foundations you should already know in order to excel in higher level courses.

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Review Chapter

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  1. Review Chapter • What you Should Learn • REALLY – WHAT YOU SHOULD HAVE ALREADY LEARNED • If not, then you might be in too high of a course level – decide soon!!!

  2. Henry David Thoreau - author • “It affords me no satisfaction to commence to spring an arch before I have got a solid foundation.”

  3. Objective • Understand the structure of algebra including language and symbols.

  4. Objective • Understand the structure of algebra including language and symbols.

  5. Definiton • Expression – a collection of constants, variables, and arithmetic symbols

  6. Definition • Inequality – two expression separated by <, <, >, >, • -2>-3 • 4 < 5 • 4 < 4

  7. Definition • Equation – two expression set equal to each other • 4x + 2 = 3x - 5

  8. Def: evaluate • When we evaluate a numerical expression, we determine the value of the expression by performing the indicated operations.

  9. Definition • Set is a collection of objects • Use capitol letters to represent • Element is one of the items of the collection • Normally use lower case letters to describe

  10. Procedure to describe sets • Listing: Write the members of a set within braces • Use commas between • Use … to mean so on and so forth • Use a sentence • Use a picture

  11. Julia Ward Howe - Poet • “The strokes of the pen need deliberation as much as the sword needs swiftness.”

  12. Examples of Sets • {1, 2, 3} • {1, 2, 3, …, 9, 10} • {1, 2, 3, … } = N = Natural numbers

  13. Set Builder Notation • {x|description} • Example {x|x is a living United States President}

  14. Def: Empty Set or Null set is the set that contains no elements • Symbolism

  15. Symbolism – element “is an element of”

  16. Def: Subset: A is a subset of B if and only if ever element of A is an element of B • Symbolism

  17. Examples of subset • {1, 2} {1, 2, 3} • {1, 2} {1, 2} • { } {1, 2, 3, … }

  18. Def: Union symbolism: A B • A union B is the set of all elements of A or all elements of B.

  19. Example of Union of sets • A = {1, 2, 3} • B = {3, 4, 5} • A B = {1, 2, 3, 4, 5}

  20. Real Numbers • Classify Real Numbers • Naturals = N • Wholes = W • Integers = J • Rationals = Q • Irrationals = H • Reals = R

  21. Def: Sets of Numbers • Natural numbers • N = {1,2,3, … } • Whole numbers • W = {0,1,2,3, … }

  22. Integers • J = {… , -3, -2, -1, 0, 1, 2, 3, …} Naturals Wholes Integers

  23. Def: Rational number • Any number that can be expressed in the form p/q where p and q are integers and q is not equal to 0. • Use Q to represent

  24. Def (2): Rational number • Any number that can be represented by a terminating or repeating decimal expansion.

  25. Examples of rational numbers • Examples: 1/5, -2/3, 0.5, 0.33333… • Write repeating decimals with a bar above • .12121212… =

  26. Def: Irrational Number • H represents the set • A non-repeating infinite decimal expansion

  27. Def: Set of Real Numbers = R • R = the union of the set of rational and irrational numbers

  28. Def: Set of Real Numbers = R • R = the union of the set of rational and irrational numbers

  29. Def: Number line • A number line is a set of points with each point associated with a real number called the coordinate of the point.

  30. Def: origin • The point whose coordinate is 0 is the origin.

  31. Definition of Opposite of opposite • For any real number a, the opposite of the opposite of a number is -(-a) = a

  32. Definition: For All

  33. Def: There exists

  34. Bill Wheeler - artist • “Good writing is clear thinking made visible.”

  35. Def: intuitiveabsolute value • The absolute value of any real number a is the distance between a and 0 on the number line

  36. Def: algebraic absolute value

  37. Calculator notes • TI-84 – APPS • ALG1PRT1 • Useful overview

  38. George Patton • “Accept challenges, so that you may feel the exhilaration of victory.”

  39. Properties of Real Numbers • Closure • Commutative • Associative • Distributive • Identities • Inverses

  40. Commutative for Addition • a + b = b + a • 2+3=3+2

  41. Commutative for Multiplication • ab = ba • 2 x 3 = 3 x 3 • 2 * 3 = 3 * 2

  42. Associativefor Addition • a + (b + c) = (a + b) + c • 2 + (3 + 4) = (2 + 3) + 4

  43. Associative for Multiplication • (ab)c = a(bc) • (2 x 3) x 4 = 2 x (3 x 4)

  44. Distributivemultiplication over addition • a(b + c) = ab + ac • 2(3 + 4) = 2 x 3 + 2 x 4 • X(Y + Z) = XY +XZ

  45. Additive Identity • a + 0 = a • 3 + 0 = 3 • X + 0 = X

  46. Multiplicative Identity • a x 1 = a • 5 x 1 = 5 • 1 x 5 = 5 • Y * 1 = Y

  47. Additive Inverse • a(1/a) = 1 where a not equal to 0 • 3(1/3) = 1

  48. George Simmel - Sociologist • “He is educated who knows how to find out what he doesn’t know.”

  49. Order to Real Numbers • Symbols for inequality • Bounded Interval notation • *** Definition of Absolute Value • Absolute Value Properties • Distance between points on # line

  50. George Simmel - Sociologist • “He is educated who knows how to find out what he doesn’t know.”

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