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Big Bend Community CollegeBeginning AlgebraMPC 095Lab NotebookBeginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond the scope of this license may be available at of ContentsModule A: Linear Equations3Module B: Graphing Linear Equations31Module C: Polynomials50Module D: Factoring72Module E: Rational Expressions93MPC 095 Module A: Linear EquationsOrder of Operations – IntroductionThe order: To remember:Example A5-3(2+42)Example B30÷52+4-72Practice APractice BOrder of Operations – ParenthesisDifferent types of parenthesis:Always do __________________________ first!Example A4+2-[52÷2+3]Example B7{22+220÷4+6}Practice APractice BOrder of Operations – FractionsWhen simplifying fractions, always simplify ___________ and ___________ first, then ____________Example A-42-4+2?35+35-4Example B4+52-923-22+3Practice APractice BOrder of Operations – Absolute ValueAbsolute Value – just like ________________, make positive _____________________Example A-3|24-5+42|Example B2-4|32+52-62|Practice APractice BSimplify Algebraic Expressions – EvaluateVariables – Dozen is ______________ as 12To Evaluate:Example A4x2-3x+2 when x=-3Example B4b2x+3y when b=-2,x=5,y=-7Practice APractice BSimplify Algebraic Expressions – Combine Like TermsTerms:Like Terms:When we have like terms we can ___________ the coefficients of _________________Example A4x3-2x2+5x3+2x-4x2-6xExample B4y-2x+5-6y+7x-9Practice APractice BSimplify Algebraic Expressions – Distributive PropertyDistributive Property:We use the distributive property to _________________________Example A-2(5x-4y+3)Example B4x(7x2-6x+1)Practice APractice BSimplify Algebraic Expressions – Distribute and CombineOrder of operations tells us that _______________ comes before ______________________So we will always ____________________ first and then _________________________ lastExample A43x-7-7(2x+1)Example B27x-3-(8x+9)Practice APractice BLinear Equations – One Step EquationsShow that x=-3 is the solution to 4x+5=-7We solve by working _____________________, using the inverse or ___________________ operations!Example Ax+5=7Example B9=x-7Example C5x=35Example Dx4=3Practice APractice BPractice CPractice DLinear Equations – Two Step EquationsWhen solving we do Order of Operations in ________________________First we will ___________ and _____________ . Then we will ________________ and _______________Example A5-7x=26Example B14=-2+4xPractice APractice BLinear Equations - GeneralMove variables to one side by ________________________________.Sometimes we may have to ________________ first.Simplify by _______________ and __________________ on each side.Example A2x+7=-5x-3Example B42x-5+3=54x-1-10xPractice APractice BLinear Equations – FractionsClear fractions by multiplying _____________ by the _________________________________Important: Multiply _______________ including _____________________Example A34x-12=56Example B35x-710=-4+715xPractice APractice BLinear Equations – Distributing with FractionsImportant: Always ________________ first and ____________________________ secondExample A12=342x-49Example B23x+4=556x-715Practice APractice BFormulas – Two Step FormulasSolving Formulas: Treat other variables like __________________.Final answer is an _____________________Example: 3x=15 and wx=yExample Awx+b=y for xExample Bab+cd=wx+y for bPractice APractice BFormulas – Multi-Step FormulasStrategy: Example Aa3x+b=by for xExample B3a+2b+5b=-2a+b for aPractice APractice BFormulas – FractionsClear fractions by __________________________________May have to _______________________ first!Example A5x+4a=bx for xExample BA=12hb+c for bPractice APractice BAbsolute Value – Two Solutions What is inside the absolute value can be ______________ or _______________This means we have ____________________Example A2x-5=7Example B7-5x=17Practice APractice BAbsolute Value – Isolate AbsoluteBefore we look at our two solutions, we must first ____________________________We do this by ______________________________Example A5+23x-4=11Example B-3-72-4x=-32Practice APractice BAbsolute Value – Two AbsolutesWith two absolutes, we need ___________________________The first equation is ___________________________The second equation is _____________________________Example A2x-6=|4x+8|Example B3x-5=|7x+2|Practice APractice BWord Problems – Number ProblemsTranslate:Is/Were/Was/Will Be:More than:Subtracted from/Less Then:Example AFive less than three times a number is nineteen. What is the number?Example BSeven more than twice a number is six less than three times the same number. What is the number?Practice APractice BWord Problems – Consecutive IntegersConsecutive Numbers:First:Second:Third:Example AFind three consecutive numbers whose sum is 543.Example BFind four consecutive integers whose sum is -222Practice APractice BWord Problems – Consecutive Even/OddConsecutive Even:First:Second:Third:Consecutive Odd:First:Second:Third:Example AFind three consecutive even integers whose sum is 84. Example BFind four consecutive odd integers whose sum is 152.Practice APractice BWord Problems – TrianglesAngles of a triangle add to ________________Example ATwo angles of a triangle are the same measure. The third angle is 30 degrees less than the first. Find the three angles.Example BThe second angle of a triangle measures twice the first. The third angle is 30 degrees more than the second. Find the three angles.Practice APractice BWord Problems – PerimeterFormula for Perimeter of a rectangle:Width is the ______________ sideExample AA rectangle is three times as long as it is wide. If the perimeter is 62 cm, what is the length?Example BThe width of a rectangle is 6 cm less than the length. If the perimeter is 52 cm, what is the width?Practice APractice BAge Problem – Variable NowTable:Equation is always for the ______________________Example ASue is five years younger than Brian. In seven years the sum of their ages will be 49 years. How old is each now?Example BMaria is ten years older than Sonia. Eight years ago Maria was three times Sonia’s age. How old is each now?Practice APractice BAge Problem – Sum NowConsider: Sum of 8…When we have the sum now, for the first box we use ______ and the second we use ______________Example AThe sum of the ages of a man and his son is 82 years. How old is each if 11 years ago, the man was twice his son’s age?Example BThe sum of the ages of a woman and her daughter is 38 years. How old is each if the woman will be triple her daughter’s age in 9 years?Practice APractice BAge Problems – Variable TimeIf we don’t know the time:Example AA man is 23 years old. His sister is 11 years old. How many years ago was the man triple his sister’s age?Example BA woman is 11 years old. Her cousin is 32 years old. How many years until her cousin is double her age?Practice APractice BMPC 095 Module B: Graphing Linear EquationsInequalities – GraphingInequalities:Less ThanLess Than or Equal ToGreater ThanGreater Than or Equal ToGraphing on Number Line – Use for less/greater than and use when its “or equal to”Example AGraph x≥-328575104140Example BGive the inequalityPractice APractice BInequalities – Interval NotationInterval notation:( , )Use for less/greater than and use when its “or equal to”∞ and -∞ always use a Example A390525264795Give Interval NotationExample B-49530255270Graph the interval (-∞,-1) Practice APractice BInequalities - SolvingSolving inequalities is just like ___________________________________The only exception is if you _______________ or ________________ by a _____________, you must__________________________________Example A7-5x≤17Example B3x+8+2>5x-20Practice APractice BInequalities - TripartiteTripartite Inequalities:When solving __________________________________When graphing _________________________________Example A2≤5x+7<22Example B5<5-4x≤13Practice APractice BGraphing and Slope – Points and LinesThe coordinate plane: Give ___________________ to a point going ______________ then _________________ as _________Example AGraph the points -2,3, 4,-1, -2,-4, 0,3, and (-1,0)Example BGraph the line: y=0.5x-2Practice APractice BGraphing and Slope – Slope from a graphSlope:Example AExample BPractice APractice BGraphing and Slope – Slope from two pointsSlope:Example AFind the slope between 7,2 and (11,4)Example BFind the slope between -2,-5 and (-17,4)Practice APractice BEquations – Slope Intercept EquationSlope-Intercept Equation:Example AGive the equation with a slope of -34 and y-intercept of 2Example BGive the equation of the graphPractice APractice BEquations – Put in Intercept FormWe may have to put an equation in intercept form.To do this we ______________________________Example AGive the slope and y-intercept5x+8y=17Example BGive the slope and y-intercepty+4=23(x-4)Practice APractice BEquations - GraphWe can graph an equation by identifying the ____________________ and _______________________Start at the _____________________ and use the ___________________________ to changeRemember slope is ________________ over _____________________Example AGraph y=-34x+2Example BGraph 3x-2y=2Practice APractice BEquations – Vertical/HorizontalVertical Lines are always ______ equals the __________Horizontal Lines are always _________ equals the __________Example AGraph y=-2Example BFind the equationPractice APractice BEquations – Point SlopePoint Slope Equation:Example AGive the equation of the line that passes through (-3,5) and has a slope of -23Example BGive the equation of the line that passes through (6,-2) and has a slope of 4. Give your final answer in slope-intercept form.Practice APractice BEquations – Given Two PointsTo find the equation of a line you must have the __________________Recall the formula for slope: Example AFind the equation of the line through (-3,-5) and (2,5).Example BFind the equation of the line through (1,-4) and (3,5). Give answer in slope-intercept form.Practice APractice BParallel and Perpendicular - SlopeParallel Lines: Perpendicular Lines:Slope: Slope:Example AOne line goes through 5,2 and (7,5). Another line goes through (-2,-6) and (0,-3). Are the lines parallel, perpendicular, or neither?Example BOne line goes through (-4,1) and (-1,3). Another line goes through (2,-1) and (6,-7). Are the lines parallel, perpendicular, or neither?Practice APractice BParallel and Perpendicular - EquationsParallel lines have the __________ slope, Perpendicular lines have ________________________ slopesOnce we know the slope and a point we can use the formula:Example AFind the equation of the line parallel to the line 2x-5y=3 that goes through the point (5,3)Example BFind the equation of the line perpendicular to line 3x+2y=5 that goes through the point (-3,-4)Practice APractice BDistance – Opposite DirectionsThe distance Table:Opposite Directions:Example ABrian and Jennifer both leave the convention at the same time traveling in opposite directions. Brian drove 35 mph and Jennifer drove 50 mph. After how much time were they 340 miles apart?Example BMaria and Tristan are 126 miles apart biking towards each other. If Maria bikes 6 mph faster than Tristan and they meet after 3 hours, how fast did each ride?Practice APractice BDistance – Catch UpA head start: _______________ the head start to his/her _____________Catch Up:Example ARaquel left the party traveling 5 mph. Four hours later Nick left to catch up with her, traveling 7 mph. How long will it take him to catch up?Example BTrey left on a trip traveling 20 mph. Julian left 2 hours later, traveling in the same direction at 30 mph. After how many hours does Julian pass Trey?Practice APractice BDistance – Total TimeConsider: Total time of 8…When we have a total time, for the first box we use ______ and the second we use ______________Example ALupe rode into the forest at 10 mph, turned around and returned by the same route traveling 15 mph. If her trip took 5 hours, how long did she travel at each rate?Example BIan went on a 230 mile trip. He started driving 45 mph. However, due to construction on the second leg of the trip, he had to slow down to 25 mph. If the trip took 6 hours, how long did he drive at each speed?Practice APractice BMPC 095 Module C: PolynomialsExponents – Product Rulea3?a2= Product Rule: am?an=Example A2x34x2(-3x)Example B5a3b7(2a9b2c4)Practice APractice BExponents – Quotient Rulea5a3= Quotient Rule: aman=Example Aa7b2a3bExample B8m7n46m5nPractice APractice BExponents – Power Rulesab3= Power of a Product: abm=ab3= Power of a Quotient: abm=a23= Power of a Power: amn=Example A5a4b3Example B5m39n42Practice APractice BExponents - Zeroa3a3= Zero Power Rule: a0=Example A5x3yz50Example B3x2y0(5x0y4)Practice APractice BExponents – Negative Exponentsa3a5= Negative Exponent Rules: a-m= 1a-m= ab-m=Example A7x-53-1yz-4Example B25a-4Practice APractice BExponents - Propertiesaman=aman=abm=abm=amn=a0=a-m=1a-m=ab-m=To simplify: Example A4x5y2z22x4y-2z34Example B 2x2y34x4y-6-2x-6y42Practice APractice BScientific Notation - Converta×10b abb positiveb negativeExample AConvert to Standard Notation5.23×105Example BConvert to Standard Notation4.25×10-4Example CConvert to Scientific Notation8150000Example CConvert to Scientific Notation0.00000245Practice APractice BPractice CPractice DScientific Notation – Close to ScientificPut number ___________________________________Then use ________________________________ on the 10’sExample A523.6×10-8Example B0.0032×105Practice APractice BScientific Notation – Multiply/DivideMultiply/Divide the ____________________________________Use _______________________________ on the 10’sExample A3.4×105(2.7×10-2)Example B5.32×1041.9×10-3Practice APractice BScientific Notation – Multiply/Divide where answer not scientificIf your final answer is not in scientific notation ______________________________________Example A6.7×10-6(5.2×10-3)Example B2.352×10-68.4×10-2Practice APractice BPolynomials - EvaluateTerm:Monomial:Binomial:Trinomial:Polynomial:Example A5x2-2x+6 when x=-2Example B-x2+2x-7 when x=4Practice APractice BPolynomials – Add/SubtractTo add polynomials:To subtract polynomials:Example A5x2-7x+9+2x2+5x-14Example B3x3-4x+7-(8x3+9x-2)Practice APractice BPolynomials – Multiply by MonomialsTo multiply a monomial by polynomial:Example A5x2(6x2-2x+5)Example B-3x4(6x3+2x-7)Practice APractice BPolynomials – Multiply by BinomialsTo multiply a binomial by a binomial: This process is often called _________ which stands for ___________________________________Example A4x-2(5x+1)Example B3x-7(2x-8)Practice APractice BPolynomials – Multiply by TrinomialsMultiplying trinomials is just like ________________ we just have ____________________________Example A2x-4(3x2-5x+1)Example B2x2-6x+1(4x2-2x-6)Practice APractice BPolynomials – Multiply Monomials and BinomialsMultiply _________________________ first, then __________________ the ___________________Example A42x-43x+1Example B3xx-6(2x+5)Practice APractice BPolynomials – Sum and Differencea+ba-b= Sum and Difference Shortcut:Example Ax+5(x-5)Example B6x-2(6x+2)Practice APractice BPolynomials – Perfect Squarea+b2= Perfect Square Shortcut:Example Ax-42Example B2x+72Practice APractice BDivision – By MonomialsLong Division Review: 5|2632Example A3x5+18x4-9x33x2Example B15a6-25a5+5a45a4Practice APractice BDivision – By PolynomialsOn division step, only focus on the _______________________Example Ax3-2x2-15x+30x+4Example B4x3-6x2+12x+82x-1Practice APractice BDivision – Missing TermsThe exponents MUST ______________________________If one is missing we will add ______________Example A3x3-50x+4x-4Example B2x3+4x2+9x+3Practice APractice BMPC 095 Module D: FactoringGCF and Grouping – Find the GCFGreatest Common Factor:On variables we use ______________________________Example AFind the Common Factor15a4+10a2-25a5Example BFind the Common Factor4a4b7-12a2b6+20ab9Practice APractice BGCF and Grouping – Factor GCFab+c= Put _______ in front, and divide. What is left goes in the _________________________Example A9x4-12x3+6x2Example B21a4b5-14a3b7+7a2b4Practice APractice BGCF and Grouping – Binomial GCFGCF can be a _____________________Example A5x2y-7+6y(2y-7)Example B3x2x+1-7(2x+1)Practice APractice BGCF and Grouping - GroupingGrouping: GCF of the ___________ and ______________ then factor out __________________ (if it matches!)Example A15xy+10y-18x-12Example B6x2+3xy+2x+yPractice APractice BGCF and Grouping – Change OrderIf binomials don’t match:Example A12a2-7b+3ab-28aExample B6xy-20+8x-15yPractice APractice BTrinomials – a≠1ax2+bx+c AC Method: Find a pair of numbers that multiply to _____ and add to _____Using FOIL, these numbers come from __ and __Example A3x2+11x+10Example B 12x2+16xy-3y2Practice APractice BTrinomials – a≠1 with GCFAlways factor the ________ first!Example A18x4-21x3-15x2Example B16x3+28x2y-30xy2Practice APractice BTrinomials – a=1If there is a ______ in front of x2, the ac method gives us ______________Example Ax2-2x-8Example Bx2+7xy-8y2Practice APractice BTrinomials – a=1 with GCFAlways do the _______ first!!Example A7x2+21x-70Example B4x4y+36x3y2+80x2y3Practice APractice BSpecial Products – Difference of Squaresa+ba-b= Difference of Squares:Example Aa2-81Example B49x2-25y2Practice APractice BSpecial Products – Sum of SquaresFactor: a2+b2Sum of Squares is always _______________Example Ax2+9Example B16a2+25b2Practice APractice BSpecial Products – Difference of 4th PowersThe square root of x4 is _____________With fourth powers we can use _____________________________ twice!Example Aa4-16Example B81x4-256Practice APractice BSpecial Products – Perfect SquaresUsing the ac method if the numbers ____________________ then it factors to __________________Example Ax2-10x+25Example B9x2+30xy+25y2Practice APractice BSpecial Products – CubesSum of Cubes:Difference of Cubes:Example Am3+125Example B8a3-27y3Practice APractice BSpecial Products - GCFAlways factor the ___________ first!!Example A8x3-18xExample B2x2y-12xy+18yPractice APractice BFactoring Strategy - StrategyAlways do ________ First2 terms:3 terms:4 terms:Example AWhich method would you use?25x2-16Example BWhich method would you use?x2-x-20Example CWhich method would you use?xy+2y+5x+10Practice APractice BPractice CPractice DPractice ESolve by Factoring – Zero Product PropertyZero Product Rule:To solve we set each ________________ equal to _________________Example A5x-12x+5=0Example B2xx-62x+3=0Practice APractice BSolve by Factoring – Need to FactorIf we have x2 and x in an equation, we need to _______________ before we ______________Example Ax2-4x-12=0Example B3x2+x-4=0Practice APractice BSolve by Factoring – Equal to ZeroBefore we factor, the equation must equal _____________.To make factoring easier, we want the ____________________ to be ____________________.Example A5x2=2x+16Example B-2x2=x-3Practice APractice BSolve by Factoring - SimplifyBefore we make the equation equal zero, we may have to ______________________ first.Example A2xx+4=3x-3Example B2x-33x+1=-8x-1Practice APractice BMPC 095 Module E: Rational ExpressionsReduce - EvaluateRational Expressions: Quotient of two ____________________________Example Ax2-2x-8x-4 when x=-4Example Bx2-x-6x2+x-12 when x=2Practice APractice BReduce – Reduce FractionsTo reduce fractions we _____________________ common ________________________Example A2415Example B4818Practice APractice BReduce - MonomialsQuotient Rule of Exponents: aman=Example A16x512x9Example B15a3b225ab5Practice APractice BReduce - PolynomialsTo reduce we _____________________ common ________________________This means we must first ___________________________Example A2x2+5x-32x2-5x+2Example B9x2-30x+259x2-25Practice APractice BMultiply and Divide - FractionsFirst _____________________ common _________________________________Then multiply _____________________________________Division is the same, with one extra step at the start: _________________ by the _______________Example A635?2110Example B58÷104Practice APractice BMultiply and Divide - MonomialsWith monomials we can use ________________________am?an= aman= Example A6x2y55x3?10x43x2y7Example B 4a5b9a4÷6ab412b2Practice APractice BMultiply and Divide - PolynomialsTo divide out factors, we must first _______________________Example Ax2+3x+24x-12?x2-5x+6x2-4Example B3x2+5x-2x2+3x+2÷6x2+x-1x2-3x-4Practice APractice BMultiply and Divide – Both at OnceTo divide:Be sure to _________________________ before ______________________Example Ax2+3x-10x2+6x+5?2x2-x-32x2+x-6÷8x+206x+15Example Bx2-1x2-x-6?2x2-x-153x2-x-4÷2x2+3x-53x2+2x-8Practice APractice BLCD - NumbersPrime Factorization:To find the LCD use ______________ factors with _______________ exponents. Example A20 and 36Example B18, 54 and 81Practice APractice BLCD - MonomialsUse _______________ factors with ___________________ exponentsExample A5x3y2 and 4x2y5Example B7ab2c and 3a3bPractice APractice BLCD - PolynomialsUse _______________ factors with ___________________ exponentsThis means we must first ________________________Example Ax2+3x-18 and x2+4x-21Example Bx2-10x+25 and x2-x-20Practice APractice BAdd and Subtract - FractionsTo add or subtract we ___________ the denominators by ________________ by the missing _____________________.Example A520+715Example B814-310Practice APractice BAdd and Subtract – Common DenominatorAdd the __________________________ and keep the __________________________When subtracting we will first _______________________ the negativeDon’t forget to ___________________Example Ax2+4xx2-2x-15+x+6x2-2x-15Example Bx2+2x2x2-9x-5-6x+52x2-9x-5Practice APractice BAdd and Subtract – Different DenominatorsTo add or subtract we ___________ the denominators by ________________ by the missing _____________________.This means we may have to ___________________ to find the LCD!Example A2xx2-9+5x2+x-6Example B2x+7x2-2x-3-3x-2x2+6x+5Practice APractice BDimensional Analysis – Convert Single UnitMultiply by ___ and value does not change1= Ask questions:Example A5 feet to metersExample B3 miles to yardsPractice APractice BDimensional Analysis – Convert Two Units“Per” is the ____________________________Clear ___________ unit at a time!Example A100 feet per second to miles per hourExample B25 miles per hour to kilometers per minutePractice APractice B ................
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