Chapter 1 - Review. Page 3 There s a chart with language that might be helpful. Let p= price. Let

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Chapter 1 Review 11 The Language of Algebra Page 3 There s a chart with language that might be helpful Ex: Translate each to a Mathematical Model (Note: Be sure to declare variables) 1) One half of

1 Chapter 1 Review 11 The Language of Algebra Page 3 There s a chart with language that might be helpful Ex: Translate each to a Mathematical Model (Note: Be sure to declare variables) 1) One half of the price p±p or 0 5p 2) The sale price is one half of the price S I Let p price Let p price 12 The Real Numbers Venn Diagrams Ex: a) What does this Venn Diagram tell us? p }d;yibie s Sale price b) Where would bearded dragons appear? How about roses?

2 ? Sets of Numbers : Natural Numbers: N {1, 2, 3 } Whole Numbers: W {0, 1, 2, 3 } W Integers: Z { 3, 2, 1, 0, 1, 2, 3 } {0, ±1, ±2, ±3 } z Rational Numbers: Q The set of numbers that can be written as a fraction, or the set of all numbers that are either terminating or repeating Terminating: Decimal 4 40 " 025 #$ Repeating: % fbrayes & ' %% : 6J Tphfi 0333 But how do we write the set in set notation? Q {, 'I, } 0 J Is 1 Q { Fla be Z, bt 0, } ftpt#sandbbezmt Squirrels Pizzas bookings as, Set builder notation for the set of Rational Numbers ( ie fractions )

3 ±: the i

4 Squirrels is Pizzas Pizzas sys I each squirrel gets I pitza Is f Each at squirrel gets pizza * 9 11 Fins Each squirrel gets 0 0 pizza 4 0

5 ? 0 squirrels 4 pizzas we don't know the answer ± undefined o 2 why? why? 0: ? ?D

6 How do you remember it? 0 undefined &E ) \ 4g death i&m0et NoPah% g

$tatkaq#g/lhuq~ (Note that some square roots are not irrational For example, ) Reals: R * lrrtakonals Trationals Union IN 0 W 1 z Q\ R µ Ex : f is an element of 4 EN MEH 4 EIW HER 4$

7 Irrational Numbers: H The set of all nonterminating, nonrepeating decimals Examples: D 14 FEZ Notlrratiap :3 :Yn I ms?tatkaq#g/lhuq~ (Note that some square roots are not irrational For example, ) Reals: R * lrrtakonals Trationals Union IN 0 W 1 z Q\ R µ Ex : f is an element of 4 EN MEH 4 EIW HER 4 EZ 4 EQ 4 E t6lt 4 IH REIN 4ER NEW a Ez of E Q f < R

8 Ex: There s one mistake in this list of Prime Numbers Find it: Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, Composite Numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 97, 98, 99, 100 Divisibility Rules: 2 s: Even number Ends in 0, 2,4, 6,8 3 s: When you add the digits of the number, the sum is divisible by 3 Ex: 57 > * A29 II " 57 n so } goesj 3 / \ 19 5 s: The number ends in 0 or 5 Ex: * 916 T 6 s: The number is divisible by 2 and T *0 9 s: When you add the digits of the number, the sum is divisible by 9 Ex: 81 > s: The number ends in 0 Pronunciation of 6: Negative six or the opposite of six What s the opposite of 6? (6) 6

9 Absolute Value Definition: The absolute value of a number is its distance from 0 on the number line Ex: Simplify 1) 6 2) Eh TT 3) 6 (6) 6 4) 0 Ex: Can you divide by 0? Why or why not? :0 The Sharks will eat you

10 13 Operations with Real Numbers BIG NOTE: Square Roots What are the square roots of 25? 25 5 Radical 25 5, The opposite of radical 25 Exponent Terminology: to the 4 th power 3 is the base 4 is the exponent On a calculator: Pa 81 3 XO 4 Enter 81 Order of Operations 8 4 * 2 0 or 8? 8 4 * 2 4 * 2 8 or Order of Operations: 1) Grouping Symbols Parentheses ( ) Brackets [ ] Braces { } Absolute Value 2) Exponents 3) Division/Multiplication, Work Left to Right 4) Subtraction/Addition, Work Left to Right

11 Ex: Simplify 1) 8 4 * ) ( ) ( ) 8180 ( 3 Fraction Review: Ex: 1) # + + qfi9qe;a II ) 8180 (341) 8180 j f 's i 2) + # t t ) # 's 8 9

12 7) c) Simplifying Algebraic Expressions Using Properties of Real Numbers Properties of Real Numbers 1) The Commutative Property of Addition a + b b + a E : Does it work for subtraction? No ± 20 l oo ) The Commutative Property of Multiplication ab ba E : X Does it work for division? 6 3 I I 3) The Associative Property of Addition (a + b) + c a + (b + c) (57+38) 4) The Associative Property of Multiplication (3 (ab)c a(bc) ( 2 1 ) ) The Distributive Property Visiting the Family a(b + c) ab + ac :( b ( ) + 38 ( 100 ) I 3 (7 8) 3 ( 56 ) 1 68 ab ac earns KEY 2 Destiny

13 a(b c) ab ac Identities Special numbers that don t change the value of other numbers 1) Additive Identity: O Ex : to 3 2) Multiplicative Identity: Ex: Simplify I Ex : 1 C2) 2 ( i 4) 114 1) # +,t6x ĒE, 2 3 yt 6x 8 3 y 2) 05[ 3 (x 4) 2 (x + 2)] 07 (x 3) 05 T F [ )] 0 5[x 2I4] ( x ( x 3) 3) 05[116] 07 ( x 3) ) ) ) Fx F To m I

14 15 Solving Linear Equations Using Properties of Equality When you re solving an equation, what steps are you allowed to use? 1) Simplifying 2x x 6 2) Adding to both sides : x 4 10 * t# 14 3) Subtracting from both sides (technically just adding the opposite) y ) Multiplying y: both sides by the same number (also known as clearing fractions) ½ x 16 zx ) Dividing both sides by the same number (again, technically this is just multiplying) 2x 16 x8

15 Ex: Solve and check 1) ante # # ' terms 3,2 LcD t 6162 t 2(3tp ) 6+2p 24p 1234>+2 ) p 6 ZZP 6 3p 6 2) 002x (110 x) y Clear decimals : new 10,000 (002A + 10,000 [ ]10,000/2175 ) 10,0000 od D Question: What do you do when an equation doesn t have a solution? Ex : xxtfi, TO No False Solution

16 Equation : 01 False No Solution ( Done Nocheckng ) Single Expression : f undefined FZ Undefined or Not Real or Imaginary

17 6 ZZP 63p +221> p p O_ 19PM 19 Check : 3 4p 2 3 4(dE 2 0 o F

18 Factors : 3p Terms : 3+p or 3 p Ex : Simplify 1 }I p Multiplication Is you I 2 3 Can't be Simplified because Is stupid addition

16 Solving Formulas; Geometry Formulas you should know by heart: Area

A D Ztr Area ]n tzbjha Perimeter,+sibz+sib3 base b Area right angle lw

(a+b)h Ex: Sixtysix feet of crown molding was installed around a

19 16 Solving Formulas; Geometry Formulas you should know by heart: Area Perimeter w, 90 Try Area Circumference (the perimeter of a circle) arg? A D Ztr Area ]n tzbjha Perimeter,+sibz+sib3 base b Area right angle lw length width Rectangle 2l+2w2( ltw ) Triangle a b " ; h Trapezoid Tr2 (a+b)h Ex: Sixtysix feet of crown molding was installed around a rectangular dining room If the length of the room is 1875 feet, what is its width? the 88IPTwiIdE ) 662W Ex: Find the area of a circle with diameter 38 ft 28 t AE' 495 "IE ff+d / 361 'T ft ' 38ft r ( 3 8) L 9ft }~~ 1134ft '

20 Edu Abh g h :# A ah too small AfCa+b)h f f just right

21 Solving for a Specified Variable Ex: Solve A P + Prt for a) t not AP+Prt t n pp APT A A t n b) P AP+Prt AP( itrt ) i ' YEH ft P '

22 P + A Solve for P Pta +6 P 317 PA +6 3Pr A + 6 Pf3r tea ) A3 P@ p A +6 ( At 6) a As

23 17 Using Equations to Solve Problems Word Problems to Review Ex: (#12, Page 71) As of October 2010, Denzel Washington s three top domestic growing films, American Gangster, Remember the Titans, and The Pelican Brief, had earned a total of $3467 million If American Gangster earned $145 million more than Remember the Titans, and if Remember the Titans earned $149 million more than The Pelican Brief, how much did each film earn as of that date? a T T smaller big Ex: (#22, Page 72) A student working for a delivery company earns $5750 per day plus $475 for each package she delivers How many deliveries must she make each day to earn $200 a day? s r Let American a Gangsters earnings ) " r Remember the Titan 's p Thepelican Brief 's 3467 Total atrtp r +145 Let D # deliveries per day " Daily wage + Deliveries Total Fixed Variables T D deliveries D 30 papcdadagfs or

24 at rtp 3467 a rt 145 r p Substitution : ft hrjfriablens a + r + p Gt 145 ) p ) + p rt Zp C Two variables in ( p ) + Zpt ~ 3p One Variable y 3p p 1008

25 r p a 145T r Totak $1008 million $1157 % million a $1302 million

26 Ex: (#36, Page 73) In the illustration, r s (read as line r is parallel to line s ), and b 137 Find a and c If 370 ) a T C C More about Problem Solving More Word Problems to Review, including Mixture Problems Ex: (#16, Page 84) Complete the following table that could be used to solve this problem: How many pints of punch from the orange cooler must be mixed with the entire contents of the blue cooler to get a 12% punch mixture? Then write an equation that could be used to solve this liquid ftpfm#" measure problem Amount * Strength Pure concentrate Strong 20 Weak Mix rpintsjo?yzoqyyzriyf+4+oiroiyzotd

27 Tge #m l 11 m l is parallel to M L ) +D A 1800 Jr 90

Chapter 1 An Introduction to Algebra

Chapter 1 An Introduction to Algebra 1.1 An Introduction to Algebra Symbols Parenthesis/Parentheses Bracket/Brackets Brace/Braces Algebraic Expressions vs. Algebraic Equations Operation Variable Constant