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Date Created: 10/20/15
MATH 582 HARGROVE CHAPTER 2 STUDY GUIDE 21 Introduction to Variables Definitions A Variables A letter ofthe that represents a B Constant A value that remains the and C Expressions A mathematical made up of and D Substitution A of a variable with a value E Coefficient The factorin front ofa Evaluating the expression Use to nd the value of an expression A Evaluate 1 Let X and y for the expression X y 2 Let a forthe expression 4a 3 Let a amp b for a j b b 4 Leta ampb for g j a b b b 5 Evaluate for X2 amp X2 forx 6 Evaluate for 2m2n3 for m and n Section 22I Simplifying Expressions Definitions A Term Each in an expression that is separated by B Like terms Terms with exactly the ll Combining like terms see the note on page 98 12x 3x 23a 4b2b 6a 3 4x22x3x 7x2 4 2a59ab3a5 7 4ab 5 2x2y322x2y32 6 azb3c5 a2b305 lll Simplifying multiplication expressions A Simplify the following 1 323X 2 84y 3 93y3 B Use the distributive property to simplify 1 24x 5y 2 65a 2 3 4 74x 7 Section 23I Solving Equations Using Addition Addition Principle or Addition Property of Eguality If then In other words you may add the same number to both side of an equation and still keep it balanced Solving Equations A ProcessGoal B Examples Solve each equation and check the solution 1 x 24 98 15298n 3n184120 4y3018 C See page 111 about checking solutions Caution D Simplifying before solving equations 1 a 7 23 2 382b9 b 24I Solving Equations Using Division Multiplication amp Division Property of Equality A The property B Using the Division Property of Equality 1 5x 200 4a 48 3 42 7y C Simplifying before solving 3X 6X 24 2 p14p218 3 3 150 139 20X 9X 4 a 96 25I Solving Equations with several Steps Solving by using the addition amp division properties A Steps 1 Use the Distributive property to remove parenthesis 2 Combine like terms on each side of the equation 3 Rewrite the equation with only one variable term 4 Rewrite the equation with only one constant term 5 Solve the equation 6 Check the solution B Solve the following equations 12x713 2 10a 911 3 2y73y1 4 a63a2 5 5c 48c5 6 8X92X9 C More dif cult problems to solve 1 3y1y2y11 2 2a39a3a33 D Solve by using the distributive property rst 1 4x 1 2 3 2c 13c7 5 35X8 93X10 7 523y1323y 2 4 5 8 20 5m 4 144a 56 2a 73n486 7n25 9 n MATH 582 HARGROVE CHAPTER 1 STUDY GUIDE INTRODUCTION TO ALGEBRA INTEGERS ThreeSte Problem Solvin Method Handout 57I MeansAverages Means The average ofa set of numbers A To nd the mean add the numbers and then divide by the number of items Mean sum of all values number of values B Find the average height of four men who measure 70 in 68 in 65 in and 69 in 1 A average height of4 men 2 A 3A C To get a B in math you must have a mean test score of at least 80 Suppose on the rst four tests your scores are 79 88 64 and 78 What is the lowest score you can get on the next test and still get a B 1 Procedure First we nd the total of ve scores needed to give us a least an 80 8O808080808050r400 Second find the total ofthe scores on your rst 4 tests 79886478309 Thus you need 400 309 91 to keep a B average 2 Using the 3step Method 2a 2b 2c 12 Introduction to Signed Numbers Positive amp negative numbers in the real world A Temperature B Altitude C Bank debits D Other cases ll How to write positive amp negative numbers A Positive numbers B Negative numbers C Types of Numbers 1 Natural or Counting Numbers 2 Whole Numbers 3 Integers lll Graphing integers on the number line Graph 43ll0 2 22 2 3 IV Comparing integers A Using the symbols gt and lt 1 gt means 2 lt means 3 One way to remember is that it will always point to the or it B Write gt or lt between each pairto make a true sentence 1 o 5 2 3 8 3 6 8 V Absolute value A Definition The absolute value ofa number is the from on the number line 1 Remember distance is always so that your answer will always be 2 Think of the number line when nding absolute value B Find the absolute value of the following 1 2 2 6 3 0 4 3 13I Adding Integers Addition with a number line A Positive Positive Positive 2 1 BPositive Negative Positive or Negative 2 1 C Negative Positive Positive or Negative 2 1 D Negative Negative Negative 2 1 lt V II Addition without a number line A 1 2 B e ore 1 2 Ceore 1 2 D 9 9 9 1 2 E Problems all mixed up 1 34 2 34 334 4 34 Addition Properties Using Integers A Commutative Property of Addition B Associative Property ofAddition C Addition Property of Zero or the Identity Property ofAddition lll Adding Groups of integers A Add all the positive numbers rst and then add all of the negative numbers Lastly add these sums together for the nal sum 1 5 2 4 839 2 Try this 30 18 13 15 4 B Another way of combining numbers Add each number as you come to it 1 5 2 4 839 2 301813154 14I Subtracting Integers Finding opposites of numbers A The opposite ofa number is called the B Find the additive inverse of each number 9 3 0 ll Subtracting integers finding opposites A or e 1 2 Be 1 2 C e e 1 2 D 99 eor 1 III Try these remember to rewrite as addition A Two numbers 1 18 3 2 18 3 3 18 3 4 183 B More than two numbers Change signs first 1 3 4 63 1 2 3 9 77 10 3 Tu this 3 4 8 10 16I Multiplying Integers Multiplication A Process 1 Iftwo factors have the same signs the product is 2 Iftwo factors have different signs the product is 3 Even number of negatives 4 Odd number of negatives B Examples of2 numbers 1 203 2 20 3 3 20 3 4 203 C More than 2 numbers 1 3gtlt 4gtltsgtlt 1gt 2 slt 2gtlt 1gt 3 1gtlt 1gtlt 1gtlt 1gtltsgt Multiplication Properties A Multiplication Property of Zero 1 2 Examples 3 530 b 28093 B Multiplication Property of1 C Commutative Property ofMultiplication D Associative Property ofMultiplication E Distributive Property 1 2 Rewrite each product using the distributive property a 59 8 b 42 7 III Application A group of scuba divers descend to 15 feet below the surface ofthe sea They continue to go deeper to reach the coral formations and they must stop every 10 feet to clear their ears What is their depth when they stop to clear their ears for the 6th time A B C 17I Dividing Integers Division of integers ab b A Rules 1 lfthe signs are the same the answers are 2 lfthe signs are different the answers are B Remember 1 a O or a is O 20aor0isequalto a 3 3 4 a C Examples 1 2 90 3 250 7 9 4 024 5 72 72 6 141 ll Combining multiplication amp division A 60 34o 5 B 6o 16 8o2 C 8104o 3 6 18I Exponents and Order of Operations 4 gt Exponents 333323 A How to read exponents 1 2 21 is read as 2 to the rst power 2 22 22 is read as 2 squared or 2 to the second power 3 202 2 23 is read as 2 cubed or 2 to the third power B Rewrite each multiplication as an exponent 1 50505050505 2 10010010010 C Simplifying expressions with exponents 1 62 2 3quot 3 4 4 5 5 52 D Be careful with negatives and exponents 1 aquot ao a a 2 a a0a0 a 3 Simplify these exponents a 24 b 24 c 42 d 42 ll Order of Operations A Rules Always follow the following order PEMDAS 1 Please 2 Excuse 3 My Dear 4 Aunt Sally B Remember 1 Always perform from left to right 2 Grouping symbols C Examples 125 1834 2 422 34l2 3 5 2 44 5252 4 162822 932 2 8 525 53 5 34 9 65 20 6 27 92 9 35 Hargrove39s Math 582 Chapter 43 Study Guide Addition amp Subtraction of Fractions 44I Adding amp Subtracting Signed Fractions Finding the LCD or LCM A The LCD or the least common denominator of two natural numbers is the that is a multiple of B Method 1 forfinding the LCD 1 List the multiples ofthe numbers 2 Look for the smallest common multiple C Examples of Method 1 1 Find the LCD for 20 amp30 2 Try this one Find the LCD for 8 amp 10 D Method 2 1 List the prime factorization for all numbers 2 Find the common prime factors 3 Multiply common and all other factors for the LCD B Examples Use prime factorization to nd the LCD forthe following numbers 1 LCD for12amp 15 2 LCD for 24 amp 36 F By variables 1 Find the LCD for 10a2bc amp 14 ab2c3 2 Find the LCD for xy amp yz ll Like denominators A When adding or subtracting like denominators add or subtract the keep the and B Examples 1 i2 2 i 2 5 5 a a 3 ix x 4 E amp 9 9 5a 5a 5a lll Adding or subtracting with unlike denominators Finding the LCD least common denominator Procedure Look at the denominators Find the LCM forthe denominator Change the fractions by finding the equivalent fractions so that they have the same denominators Add or subtract and simplify Finding the LCD and solving 1 Add 31 2 22 9 12 6 9 31 4 3 10 25 5 7 24x 36 IV Solving Problems A A recipe for fudge brownies calls for cup ofoil and g cup of milk How many cups of liquid ingredients are in the recipe B Celeste is replacing a 2in thick shelf in her bookcase If her replacement board is E in thick how much should be planed down before the repair can be completed 45I Problem Solving Addition amp Subtraction of Mixed Numbers Addition amp Subtraction A Method 1 Add whole numbers amp add fractions 1 5 62 2 9 32 8 6 4 6 B Method 2 Change to improper fraction and add 5 5 5 6 8 6 C Subtraction Without borrowing 9 2 D Subtraction V th borrowing 1 71 21 3 14 93 6 4 5 ll Application A An Lshaped room consists ofa rectangle that is 8 ft by 11 ft adjacent to one that is 6 ft by 7ft What is the total area ofa carpet that covers the oor see diagram 8ft 1 2 5 r 11 7ft B One day the stock of Intel Corporation opened at 1002 and then rose 4 Find the price of the stock at the end of the day C Recently in college football the distance between goalposts was reduced from 23ft to 18 ft By how much was it reduced 46I ExponentsI Order of Operations and Complex Fractions Exponents amp Fractions A Remember that 1 32 3 3 2 B Simplify the following 1 3 5 A Order of Operations Remember PEMDAS 1 Simplify 3 Simplify lZi 3821 2l3 5 6 5 32 3o3 3 fxoxox 2 l 3 2 l i 2 3 Complex Fractions are fractions in which the numerator and or denominator contain A Examples Sslqb la B Simplify the following complex fractions 5 3 1 2 2 E 10 4 47I Problem Solving Equations Containing Fractions Multiplication amp Division Properties of Equality A For any numbers a b amp c with Ca 0 If Then B Solve the following 1 3x15 23a i 4 7 3 Rix 4 zyzi 3 9 7 Using the Addition and Multiplication Properties of Equality A Addition Property of Equality lfa b then B Solve each equation 4 2 8 4 3 p 1 a610 4 43 4 54y 5 2x 2 4Z 5 5 Hor gr ove39s MATH 582 Chapter 6 Ratio and Proportion 61I Ratios l Ratios A A ratio is the quotient of 2 quantities and compares two that have the same B Every year the average American drinks 183 gallons of liquid Of this 21 gallons are milk The ratio of milk to the total amount drunk is shown as or C Finding Ratios 1 Write fractional notation for the ratio 5 to 7 2 Find the ratio of 243 to 100 1 2 3 Find the ratio of 52 to 6g 4 For every 6 Americans who use a fastfood restaurant s drivethrough window 4 others order indoors What is the ratio ofdrivethrough orders to indoor orders 5 I a 6 For every dollar spent on food about 13 cents goes to pay for the package What s the ratio ofthe cost ofthe package to the cost ofthe package s contents ll Simplifying Notation for Ratios A Simplify by B Try these 1 65 days to 45 days 2 5 t06 3 g 4 26 inches to3feet 62I Rates amp Unit Prices Rt a es A A rate compares two kinds of measureunits 1 Always include the because they do not dividecancel 2 Comparison words are B Examples 1 Mike s moped travels 145 miles on 25 gallons of gas How many miles per gallon does he get 2 A cook buys 10 lb of potatoes for 369 What is the rate he spends in cents per pound 3 What is the rate if 500 miles is traveled in 5 hours C Please note 1 Dollars to cents 2 Cents to dollars 3 0723 7232 not 07232 4 ll Unit Pricing or Cost per Unit A Unit price or unit rate is the ratio of to the and has in the denominator B lfyou buy a 40 lb bag of Science Diet dog food for 36 what is the unit price in dollars per pound C Find the unit rate 1 435 for 3 pounds of cheese 2 95 gallons of gas for 2365 63I Proportions l Proportions A When two sets of numbers have the ratio they are said to be w More examples 1 2t05amp4to7 21lampl to 1amp3 4 2 5 Solving Proportions A Use cross multiplication to solve for a missing variable BExamples 1 L 39 8 5 154 22 27 1l 87 3 4 4 L a 3g 11 04 64I Applications of Proportions A When solving proportions use the following 3step method 1 X variable statement 2 Equation will be written as a ratio equaling another ratio and it will be labeled MZM 3 Solution X yds in in al al B Make measurements equal or sec sec m1 m1 1 The weight of water is 62 lb per cubic foot What is the weight of 5 cubic feet of water 2 The tape in an audio cassette is played at the rate oflg in per second How many inches oftape are used when played for 5 seconds 3 An 8 lb turkey breast contains 36 servings of meat How many pounds ofturkey breast would be needed for 54 servings C I use proportions to find out your grades for your lab and your notebook 1 Your lab is worth 25 points However there are 50 problems in your lab and you only get 45 problems correct How many points will your earn 2 Your notebook is worth 25 points You forget to correct your test so you loose 5 points What is your score on a scale of 100 for your notebook CHAPTER 7 PERCENT NOTATION 71 Basics of Percent Understanding Percent Notation A n means n per hundred B Percent notation can be expressed in three ways 1 Fractional notation 2 Decimal Notation Converting from percent notation to a decimal A Converting from percent notation to a decimal 1 5 2 245 3 112 B Second way to convert from a percent to a decimal 0 2 245 3 112 C Converting from a decimal to a percent 1 045 2 0325 3 06 D Second way to convert from a decimal to a percent 1 045 2 0325 3 06 Converting between fractional notation and percent notation A By division 3 2 1 2 5 3 3 3 3 42 8 4 B From percent to fraction 1 36 2 112 3 112 4m3 3 73I Solving Percent Problems Using Eguations n of b is a II Translating amp Solving Equations A What is b of n or b of n is what number 1 What is11 of48 2 40 of 60 is what number B n ofwhat no is a or a is n of what no 1 120 ofwhat number is 60 2 15 is 30 ofwhat number C What percent of b is a or a is what percent of b 1 10 is what percent of 20 2 What percent of 50 is 7 I Try these A What is 87 of 76000 B 12 is what percent of 40 74I Problem Solving with Percent Applied problems involving percent A In a recent year the United States generated 815 million tons of paper waste of which about 326 million tons were recycled What percent of paper waste was recycled B The U 8 Postal Service estimates that we read 78 of the junk mail we receive Suppose that a business sends out 9500 advertising brochures How many can the business expect to be opened and read Percent of Increase or Decrease A To find a percent of increase or decrease ask the question What percent of the original amount is the increase or decrease B Find the increasedecrease 100 452 675 105 370 C V th proper furnace maintenance a family that pays a monthly fuel bill of 78 can reduce their bill to 7020 What is the percent of decrease D According to USA Today the median Sacramento Kings salary was 5 million in 20052006 and decreased to 44 million in 20062007 What is the percent of decrease 75I Consumer Application Sales TaxI Discount amp Commission I Sales Tax A Formulas 1Total price Purchase price Sales Tax Sales Tax Rate 0 Purchase Price 2Total price Purchase price Sales Tax Rate 0 Purchase Price B In 2007 the sales tax rate in Placer County California was 725 How much tax was charged on the purchase of3 CD s at 1695 each What was the total price C The sales tax is 32 on the purchase of an 800 sofa What is the sales tax rate D The sales tax on a laser printer is 3174 and the sales tax rate is 82 Find the purchase price before taxes are added Discounts A Formula Sale price Originalecount Discount rate of discount 0 original price B A rug marked 240 is on sale at 25 off What is the discount What is the sale price C A Ford 350 truck normally sells for 50000 It is on sale for 30 offand the sales tax is 725 What is the new sale price What is the total price lll Commission A Formula Commission commission rate 0 sales B A salesperson s commission rate is 20 What is the commission from the sale of 25560 worth of stereo equipment C Dawn earns a commission of 3000 selling 60000 worth of farm machinery What is the commission rate D Jan s commission rate is 25 She receives a commission of 1 825 on the sale of a scooter How much did the scooter cost
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