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Class Work #1 – M105 Fall 2011 Due: Thursday, September 29 Adding/Subtracting Decimals Line up decimal places and add/subtract as normal, bringing down the decimal as it is crossed Ex 1: 542 237.379 Multiply Decimals Multiply #’s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left Ex 2: (0.08)(0.0009) Ex 3: (1.5)2 Dividing Decimals Quotient (answer to ÷ prob), Dividend (the # being subdivided) & Divisor (# of equal parts dividend is to be subdivided into) Decimal always placed after ones in dividend, zeros can be added indefinitely to the right Terminating Decimal (an exact answer is achieved) Repeating Non-Terminating Decimal (a bar is used over the repeat to show, never use fractions in a decimal, don’t round unless specifically asked or need to for a real-world application) Non-Repeating Non-Terminating Decimals are irrational numbers and can’t be achieved by dividing one # by another Division by decimal: Move decimal out of divisor, moving it the same number of places (to the right) in the dividend Ex 4: 5 ÷ 0.25 Ex 5: 5.2 ÷ 0.3 Ex 6: 277 ÷ 80 Multiplication Property of Zero Anything times zero is zero: a•0=0 Ex 7: (-7.88)(0) Set Notation Elements or members of a set are the things that can belong to the set Roster Form is a list of elements using braces, a list, possibly ellipsis notation to indicate a pattern Set Builder Notation is a description of the elements using braces, a variable to describe the elements, a bar “|” to indicate such that the elements are described as “blah”, a description using mathematical symbols and or English words to describe Sets of Numbers A group or collection of things (elements or members); In math numbers form sets Real Numbers ( ) Natural Numbers (N): called counting too, {1, 2, 3, …}, subset of real, whole, integers, rationals Whole Numbers (W): natural #’s plus zero, {0, 1, 2, …}, subset of real, integers, rationals Integers (Z): counting #’s & opposites and zero, {…, -2, -1, 0, 1, 2, …}, can be subdivided into positive (whole numbers) and negative integers; subset of reals & rationals Rational Numbers (Q): quotient of integers where divisor ≠ 0, can’t be described in roster, {x | x ∈ p/q, q ≠ 0},subset of reals; mutually exclusive of irrationals Irrational Numbers (I): any real that’s not rational, no roster possible, {x | x p/q, q ≠ 0}, subset of reals; mutually exclusive of rationals Subset is a set contained within another set Ex 8: . List the numbers in the set that follows that are from the given sets: {-9, -1.25, 0, 3/4, √3, π, 3.3333, √25, 12 1/2, 98} a) Natural #’s b) Whole # c) Integers d) Rational #’s e) Irrational #’s Y. Butterworth Ch. 1 Concepts Review – Int. Alg. FTHL 1 Union & Intersection Union is a mathematical “or”; Symbol ∪; It is collects everything in all sets Intersection is mathematical “and”; Symbol ∩; It is what is in common in all sets When there are no elements in a set we call it the empty set or null set and use { } or Ex 9: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} & B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} & C = {1, 3, 5, 7, 9, 11} a) Give the set A ∪ B b) Give the set A ∩ B c) Give the set A ∩ C Absolute Value The distance from zero regardless of direction (the number w/out its sign) Absolute values DON’T “distribute”; they are grouping symbols – do problem inside and then take abs. val Comparison of Numbers < is less than, > is greater than, ≤ is less than or equal to & ≥ is greater than or equal to How to tell apart: Old way is little eats big, new way is point out the small guy Order Property of Real #’s: On # line left gets smaller & right gets bigger (neg. are all smaller than pos.) Decimals: Compare number by number, find the smaller, you’ve found the smaller Fractions: Cross mult. up (denom to num) and the larger product is larger fraction Neg. #’s: Larger the number looks without it’s sign the smaller it is! Ex 10: Use <, > or = to compare. Get 2 numbers to compare where needed. a) | - 15 | - ( - 15) b) | 43,472 | - | - 43,472 | c) 0.25 0.251 d) 5/12 e) -2.1 -2.2 f) -3/11 7 /17 -4/15 Adding Fractions/Mixed #’s Must have an LCD to add/subtract Find LCD by prime factorization and unique primes to highest exponent (find product) Build higher terms by using Fundamental Theorem of Fractions (mult. old denom by constant to get LCD & mult. old num by same constant to get new num) All fractional answer are in lowest terms/reduce. Fundamental Thm of Fractions to divide out GCF Improper Fractions should be changed to mixed #’s Ex 11: Fractions must be used to complete fraction problems. Work shown. 1 a) /6 + 3/10 b) 542 13/36 237 23/24 Multiplying Integers + • + = +, – • – = +, – • + = – or + • – = – Addition of Real Numbers (Integer Addition) Subtraction is not allowed, change to addition by adding the opposite of the number following the subtraction symbol Like Signs when adding the numbers add and you keep the sign Unlike Signs when adding the big minus the small and bigger #’s sign is sign of answer Ex 12: a) -13 Ex 13: Ex 14: Change each problem to addition – -29 b) -19 – 13 c) 13 – -19 Solve each addition problem created in Ex 12 5 /6 1 1/3 Y. Butterworth Ch. 1 Concepts Review – Int. Alg. FTHL 2 Order of Operations PEMDAS – Parentheses, Exponents, Mult/Division (left to right order), Add/Subt. (left to right order) Parentheses is generic for grouping symbols which include parentheses, brackets, braces, absolute values, radicals, fractions bars Most common errors: add/subt. before mult. divide & mult. before dividing Ex 15: Simplify using strict order of operations. Show work in each step. a) 24 ÷ 6 2 + 2 b) 2 8 23 4 + 2(13 10 + 5)2 c) -2 | 6 27 | 3 + 4 16 32 23 Multiplying Fractions Cancel if possible (dividing out a common factor from num & denom) Mult. numerators & mult. denominators If improper change to mixed number Check for common factors, especially if you didn’t try to cancel Ex 16: Use fractions to multiply a) ( 3/16 )( 4/7 ) b) (1 1/3)2 Dividing Fractions/Mixed #’s Change all mixed Multiply the dividend by the reciprocal of the divisor (dividend ÷ divisor = quotient!!) Follow multiplication process Ex 17: Use fractions to divide a) 4/30 5/14 b) 5 ÷ 2 2/3 Exponents Represents repeated multiplication (base used as a factor number of times indicated by exponent) Exponents only apply to the number that they are written above & to the right of Fractions: numerator & denominator to exponent (if lowest terms to start will be in lowest terms in end) Decimals: see mult. decimals Grouping Symbols: Simplify inside 1st then take single number to power One to any power is one: 1n = 1 Negative number to even power (parentheses around the negative #) is always positive Neg. # to odd power (parentheses around neg. #) is always negative -an ≠ (-a)n when n is even -an is read as: The opposite of a to the nth power (-a)n is read as: A negative number used as a factor n times Anything to the zero power is one: a0 = 1 Ex 18: Expand and simplify. a) c) 2 –1 d) (-2xy)0 g) (2xy) -1 (-3)2 e) b) 2xy0 - 32 f) 2xy -1 Evaluation Put in the values given for the variables, using parentheses to replace the variables with the values Simplify using order of operations Distributive property should never be used in lieu of order of operations This is taught as a first step in a check for an equation, and it’s use in many solution methods Ex 19: Evaluate a) x + y – z b) |x + z| – y if x = 2, y= 1 & z = -5 Ex 20: Evaluate Y. Butterworth /y + z – y/x x if x = 2, y = 5 & z = 7/30 Ch. 1 Concepts Review – Int. Alg. FTHL 3 Properties of the Real Numbers Commutative Property of Addition and of Multiplication (move addend/factors around) Associate Property of Addition and of Multiplication (group addends/factors in different orders) Identity Property of Addition and of Multiplication (gives back the identity using identity element) Identity Element of Addition: ZERO Identity Element of Multiplication: ONE Inverse Property of Addition and of Multiplication (inverse is used to give back identity element) Distributive Property (Multiplication distributes over add/subt) Division by Zero: UNDEFINED Zero Divided by Anything: ZERO (division is multiplication by a recip so becomes zero times anything) Multiplication by Zero: ZERO Ex 21: Simplify a) 0 -251 b) 502 0 Ex 22: Name the property used in each of the following a) y (5 2) = (5 2)y b) (2 + 3) + 8 = 8 + (2 + 3) c) -6 1 = -6 d) 5 + -5 = 0 e) ¼ (2 + 4z) = ½ + z Translation There is a whole separate sheet with all the translation nuances Ex 23: Translate each of the following into expressions or equations. If there is an unknown number let that number be x. Do not simplify. a) The difference of some number and the product of fifteen and the sum of the number and negative 9. b) The quotient of nine subtracted from some number and 27. c) Twenty-three less some number is equivalent to five more than twice that number. d) Three less than some number is divided by the product of the number and nine. Simplifying Algebraic Expressions Expressions have no equal sign; they can’t be solved; they can only be evaluated &/or simplified Terms are parts of an algebraic expression separated by addition or subtraction; parentheses are like presents – you can’t see inside them to see any terms (use distributive property to see terms) Like terms are determined by the variable portion being EXACTLY the same Numeric coefficients are the number multiplied by a variable(s) 1st use distributive property to get individual terms Physically or “mentally” use commutative & associative properties to group like terms Add numeric coefficients of like terms & rewrite as a “sum” of unlike terms 1 Ex 24: Simplify /3 + 2/5(x 1/3) Solving Equations in 1 Variable Equations are the equality of 2 expressions Equations can be solved to find a value that creates a true statement Solutions to an equation can be checked through evaluation Simplify: 1) Distribute 1st 2) Clear fractions/decimals 3) Combine like terms Addition Property of Equality used to move things that are add/subt. from one another across the equal sign (can be used twice) Multiplication Property of Equality used to remove numeric coefficient of variable (last step used only once) Give answers as x = #, or as a solution set in roster form 3 Types of Equations: 1) Conditional 2) Identity 3) Contradictions 3 Types of Solutions from 3 Types of Eq.: 1) Single Solution 2) All Real Numbers 3) No Solution Ex 25: Solve a) 2(x 5) + 3x = 2 3(x + 4) 1 2 c) /3 + /5(x 1/3) = 1/10 x + 3/5 e) 5(x 4) + 4x = 9(x 7) + 43 Y. Butterworth b) 2x + 5 = 2(x + 5) d) 9x 6(x 2) = 2(x + 6) f) 5½ = x 9 ¾ Ch. 1 Concepts Review – Int. Alg. FTHL 4 Solving an Equation for 1 Variable Follow the process for solving an equation, only focusing on the variable of interest h = g 7e 4 -12 + 2y = 3x Ex26: Solve the following for “e” Ex 27: Solve the following for “y” Percentage Problems Percentage to decimal conversion: Move the decimal 2 places left (remember that decimal always comes after ones position) Decimal to Fraction Conversion: Read the decimal and write what you read or count the number of decimal places and put the number in the decimal over a factor of 10 with the number of decimal places that you just counted Set up as algebra problem: _____% of ______ (whole) is ________(part) where percent as a decimal is multiplied by the whole and is equal to the part Set up as a proportion: is over of equals some part of one hundred Simple Interest: PRT = I % Increase/Decrease Problems: Original Price (op) is unknown and % is a known, and final result is known (price after increase or decrease) _____% of ________ (op) is _______ (increase/decrease) and then an equation results: op ± increase/decrease = price after which can be solved for op Ex 28: Convert each to the other 2 equivalent forms. Every answer must be exact and in the lowest terms where needed. a) 1/9 b) 0.375 c) 225% d) 66 2/5% e) 1.222 Ex 29: Solve the following problems. Give ___% of ___ is ___ and then use algebra to solve the problem. Conversions of percentages to decimals or fractions and algebra need to be shown. a) What is 40% of 55? b) 80 is what percent of 120? c) 60% of what number is 99? Ex 30: Malory goes to the store and finds a 40% off sale on her favorite dishes. If Malory pays $75 for the set, what would Malory pay for the dishes if there was no sale? Algebra must be used, and work using decimals or fractions needs to be shown. Radical Basics A radical finds the base when presented with the answer and the exponent In Other Words: A square root “undoes” a square Radicand is under the radical sign(original answer), Index is number outside to left or radical(original exponent; missing when original exponent was 2), Answer is root(original base); EG Original 22 = 4 Radical √4 = 2 If an index is even and set of numbers drawn from is real #’s, then radicand can’t be negative or root is No Solution If index is odd and radicand is negative then the root is negative Ex 31: Give the real root 3 4 a) -64 b) -81 Y. Butterworth c) 16 121 Ch. 1 Concepts Review – Int. Alg. FTHL 5 Word Problems Read, Understand, Write Basic Info in Phrase (no #’s at 1st), Form Mathematical Plan w/ Operators & Words/Phrases, Fill in #’s & Variables & Form Expressions, Substitute #’s, Variables & Expressions into Mathematical Equation/Expression, Solve, Answer w/ appropriate units Ex 32: The height of Mt. Shasta is 14,162 ft above sea level and Death Valley is 282 ft below sea level (a negative number). What is the difference in the altitude between Mt. Shasta and Death Valley? Ex 33: The Ringers play 5 more games than the Setters during a regular season. If together they play a total of 51 games how many does each team play? This is not all the concepts in Chapter 1, but this is an adequate review. I do not have time to cover every concept in the detail that I would like. I will leave it to you to review on your own and to look over my supplementary notes (Ch. 1 on my web page). Please do not put the review on the back burner, for it may become very important sometime in the very near future! Y. Butterworth Ch. 1 Concepts Review – Int. Alg. FTHL 6