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Chapter One

Chapter One. Properties of Real Numbers. 1.1 ORDER OF OPERATIONS. P ARENTHESES (GROUPING SYMBOLS) E XPONENTS M ULTIPLICATION AND D IVISION A DDITION AND S UBTRACTION *PERFORM THESE OPERATIONS AS THEY OCCUR FROM LEFT TO RIGHT. PLEASE EXCUSE MY DEAR AUNT SALLY. Order of Operations.

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Chapter One

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  1. Chapter One Properties of Real Numbers

  2. 1.1 ORDER OF OPERATIONS • PARENTHESES (GROUPING SYMBOLS) • EXPONENTS • MULTIPLICATION AND DIVISION • ADDITION AND SUBTRACTION *PERFORM THESE OPERATIONS AS THEY OCCUR FROM LEFT TO RIGHT PLEASE EXCUSE MY DEAR AUNT SALLY

  3. Order of Operations • Simplify: • [384-3(7-2)3]/3 • 3 • S-T(s2-t) if s=2 and t=3.4 • -0.04 • 8xy+z3 y2+5 if x = 5, y = -2, z=-1 • -9

  4. Order of Operations • Find the area of a trapezoid with base lengths of 13 meters and 25 meters and a height of 8 meters. • A = ½*h*(b1+b2) • A = ½*8*(25+13) • A = 152m2

  5. 1.2 REAL NUMBERS • RATIONAL #S (Q) • M/N; where N is not 0 • FRACTION, TERMINATING, REPEATING • INTEGERS (Z) …-2, -1, 0, 1, 2, … • WHOLE #S (W) 0, 1, 2, … • NATURAL #S (N) 1, 2, 3, … • IRRATIONAL #S (I) • NOT RATIONAL, NON-TERMINATING, NON-REPEATING • Examples: • Pi, .010001001000001023..

  6. Real Number Venn Diagram Naturals

  7. NAME THE SETS OF #S TO WHICH EACH BELONG Q, R Q, R I, R Q, R N, W,Z,Q,R

  8. PROPERTIES of Real Numbers • COMMUTATIVE (2 + 3) + 4 = 4 + (2 + 3) • ASSOCIATIVE - 2 (3X) = (- 2∙3) X • IDENTITY a + 0 = 0 + a • INVERSE • ADDITIVE INVERSE OF 7 IS -7 • MUSTIPLICATIVE INVERSE OF 7 IS 1/7 • DISTRIBUTIVE

  9. Distributive Property • a(b + c)=ab + ac and (b + c)a=ba + ca • Like terms have the same variables and same exponents • Ex: Simplify 4(3a – b) + 2(b + 3a)

  10. 1.3 Verbal Expressions • Three more than a number • X+3 • Six times the cube of a number • 6X3 • The square of a number decreased by the product of 5 and the same number • X2-5X • Twice the difference of a number and six • 2(X-6)

  11. Verbal Expressions • 14+9=23 • The sum of 14 and 9 is 23. • 6=-5+X • Six is equal to -5 plus a number • 7Y-2=19 • Seven times a number minus 2 is 9.

  12. Properties of Equality • Reflexive • a=a • Symmetric • If a =b, then b = a • Transitive • If a = b and b = c, then a = c • Substitution • If a = b, then a may be replaced by b and b may be replaced by a.

  13. Solving One Step Equations • S-5.48=0.02 • S=5.5 • Make sure to check solution! • 18= 1 t 2 • T=36

  14. Solving Multi Step Equations • 2(2x+3)-3(4x-5)=22 • X=-1/8 • 53=3(y-2)-2(3y-1) • X = -19

  15. Solve for a Variable • S=∏RL + ∏R2; Solve for L • L =S- ∏R2 ∏R • Re-write the formula for area of a trapezoid for h. • A = ½*h*(b1+b2) • H= 2A . b1+b2

  16. More equations…. • If 3n-8=9/5, what is the value of 3n-3? • What are two ways we can solve this problem? • Solve for n and then plug into second equation • Make left hand side look like 3n-3 • How do we do that? • Add Five to both sides • Answer = 34/5 • If 4g+5=4/9, what is the value of 4g-2? • Subtract 7 from both sides • Answer = -59/9

  17. More equations…. • Josh and Pam have bought an older home that needs some repair. After budgeting a total of $1685 for home improvements, they started by spending $425 on small improvements. They would like to replace six interior doors next. What is the maximum amount they can afford to spend on each door? • Let C represent the cost to replace each door. • 6c+425=1685 • They can spend $210 on each door.

  18. More equations…. • Basketball Game

  19. Warm Up • 1. Draw the Venn Diagram we used to review the relationship of the different sets of numbers (R, I, W, Q, Z,N) • 2. Reflect on one (or more) concepts that you feel you need to review before the quiz tomorrow.

  20. 1.4 Solving Absolute Values • Absolute Value • For any real number a, if a is positive or zero, the absolute value of a is a. If a is negative, the absolute value of a is the opposite of a. • For any real number a, |a| = a, if a ≥ 0, |a| = -a if a <0. • |3|=3 and |-3|=3 • Number Line

  21. 1.4 Solving Absolute Values • Solve 2.7 + |6-2x | if x = 4. • X=4.7 • Solve |x-18|=5. • Case 1 a = b • X-18=5 • X=23 • Case 2 a = -b • X-18=-5 • X=13 • Solution Set • {13,23}

  22. 1.4 Solving Absolute Values • Solve |y+3|=8 • (-11,5) • Solve |5x-6|+9=0 • |5x-6|=-9 • No Solution

  23. 1.4 Solving Absolute Values • Solve |x+6|=3x-2 • Case 1 • a = bx+6=3x-2 • X=4 • Case 2 • A=-bx+6=-(3x-2) • X=-1 • Double Check both solutions • Only x=4 works

  24. 1.4 Solving Absolute Values • Solve |8+y |=2y-3 • Y = 11 • 4 |3t+8 |=16t • T=8 • 3|2a+7 |=3a + 12 • A = (-11/3, -3) • -12|9x+1|=144 • No Solution

  25. 9.1 Warm Up • Grab Calculator • NO NEED FOR WARM UP BOOK • Compare homework answers with neighbor • Are you ready for the quiz? Do you have specific questions to review? Gather your thoughts and be ready to review for/take quiz.

  26. 1.5 Solving Inequalities • Trichotomy Property • Property of Order • A<B, A=B, OR A>B; only ONE of these is true • What happens if I add the same thing to both sides? Is the same inequality still true? • A+C<B+C Yes still true!

  27. Solving Inequalities • 4y-3<5y+2 • {y|y>-5} • Show on Number Line • 2(4t+9)≤18 • {t|t ≤0} • Show on Number Line

  28. Solving Inequalities • 12 ≥ -0.3P • {p|p≥-40} • Show on Number Line • 2(g+4)<3g-2(g-5) • {g|g <2} • Show on Number Line

  29. 9.2 Warm up • Get Calculator • Compare HW answers with partner

  30. Solving Inequalities Parenthesis are always used with the symbols -∞ and +∞ because endpoints can not be included here. Parenthesis is used to indicate that the endpoint is NOT included Bracket is used to indicate that the endpoint IS included • Interval Notation • Uses the infinity symbol (-∞, +∞) to indicate the unbounded interval in the positive or negative direction. • X<2 Number Line • (- ∞,2) • X ≥ 2 Number Line • [-2,+ ∞)

  31. Solving Inequalities • {m|m <7/3} • Number Line • (- ∞,7/3) [-2/5,+ ∞) Number Line

  32. Inequalities • Craig is delivering boxes of paper to each floor of an office building. Each box weighs 64 pounds, and Craig weighs 160 pounds. If the maximum capactiy of the elevator is 2000 pounds, how many boxes can Craig safely take on each elevator trip? • Formula? • 160+64b ≤ 2000 • b ≤28.75 • He can only take 28 boxes because he can not take a fraction of a box.

  33. 9.3 Warm Up • Mrs. Innerst has at most $10.50 to spend on candy for her class. She buys a soda and candy bar for herself and spends $1.55. If each piece of candy is $1.35, how many pieces of candy can she buy? • Formula? • 10.50 ≤1.55+1.35x • 6 pieces

  34. 1.6 Solving Compound and Absolute Value Inequalities When it is and, the intersection of the solution sets of the two inequalities is the answer • Compound Inequality • Consists of two inequalities joined by the word and or the word or

  35. Compound Inequalities When it is and, the intersection of the solution sets of the two inequalities is the answer • Compound “And” Inequality

  36. Compound Inequalities • 13<2x+7≤17 • Two Ways to solve • Break Apart • 13<2x+7 AND 2x+7≤17 • Solve separately and then add back together • 3<x and x ≤5 • 3< x ≤5 • Keep Together • Solve at the same time 13<2x+7≤17 • Same answer 3< x ≤5 • Plot on Number Line

  37. Compound Inequalities When it is or, the union of the solution sets of the two inequalities is the answer • Compound “OR” Inequality

  38. Compound Inequalities • Y-2>-3 or y+4 ≤-3 • Solve Separately • Y-2>-3 • Y>-1 • y+4 ≤-3 • y ≤-7 • Write Complete answer as y>-1 OR y ≤-7

  39. Compound Inequalities • 10≤3y-2<19 • {y|4≤y<17} • X+3<2 or -X ≤-4 • {X|X<-1 or x≥4}

  40. Warm Up 9.7.10 • Grab Worksheet from front and work on Numbers 11-14 from Part 1

  41. Absolute Value Inequalities • |a|<4 • What does this mean? • The distance between a and 0 on a number line is less than 4 units. • Show on number line • {a|-4<a<4} • Looks like an AND inequality

  42. Absolute Value Inequalities • |a|>4 • What does this mean? • The distance between 0 and a is GREATER than 4. • Show on number line • {a|a>4 or a <-4} • Looks like an OR inequality

  43. Absolute Value Inequalities • Summary: • |a|<b -b<a<b • |a|>ba>b or a<-b

  44. Absolute Value Inequalities • |3x-12 |≥6 • Is this an and or an or equation? • 3x-12 ≥6 OR 3x-12≤-6 • {x |x ≥6 or x ≤2}, number line • |3w+2| ≤5 • Is this an and or an or equation? • -5≤ 3w+2 ≤5 (AND) • {w |-7/3 ≤ w≤1}, number line

  45. Absolute Value Inequalities • According to a recent survey, the average monthly rent for a one bedroom apartment in one city is $750. However, the actual rent for any given one bedroom apartment might vary as much as $250 from that average. • Write an absolute value inequality to describe this situation. • |750-r | ≤250 • {r | 500≤r ≤1000} • What is the range of prices you may find in this city? • $500 to $1000

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