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Understanding the Properties of Real Numbers and Order of Operations

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This comprehensive guide delves into the essential properties of real numbers and the order of operations, focusing on grouping symbols, exponents, and performing arithmetic operations. Explore the concepts of rational and irrational numbers, along with various examples, including simplification and area calculation of geometric shapes. Gain insights into verbal expressions and the properties of equality, followed by techniques to solve one-step and multi-step equations, as well as absolute value equations. This resource is ideal for students seeking a solid foundation in mathematical principles.

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Understanding the Properties of Real Numbers and Order of Operations

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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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