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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. 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

  19. 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) • Now show on Number line

  20. 1.4 Solving Absolute Values • Solve |y+3|=8 • (-11,5) • Solve |5x-6|+9=0 • |5x-6|=-9 • No Solution

  21. 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

  22. 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

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