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Chapter 1BasicsAlgebraGraphing with CoordinatesGraphs in two dimensions are very common in algebra and are one of the most commonalgebra applications in real life.yCoordinatesThe plane of points that can be graphed in 2 dimensions iscalled the Rectangular Coordinate Plane or the CartesianCoordinate Plane (named after the French mathematicianand philosopher René Descartes).Quadrant 2Quadrant 1xQuadrant 3Quadrant 4 Two axes are defined (usually called the x‐ and y‐axes). Each point on the plane has an x value and a y value, written as: (x‐value, y‐value) The point (0, 0) is called the origin, and is usually denoted with the letter “O”. The axes break the plane into 4 quadrants, as shown above. They begin with Quadrant 1where x and y are both positive and increase numerically in a counter‐clockwise fashion.Plotting Points on the PlaneWhen plotting points, the x‐value determines how far right (positive) or left (negative) of the origin the point isplotted. The y‐value determines how far up (positive) or down (negative) from the origin the point isplotted.Examples:The following points are plotted in the figure tothe right:A (2, 3)B (‐3, 2)C (‐2, ‐2)D (4, ‐1)O (0, 0)Version 3.4in Quadrant 1in Quadrant 2in Quadrant 3in Quadrant 4is not in any quadrantPage 10 of 187April 6, 2022

Chapter 1BasicsAlgebraLinear PatternsRecognizing Linear PatternsThe first step to recognizing a pattern is to arrange a set of numbers in a table. The table canbe either horizontal or vertical. Here, we consider the pattern in a horizontal format. Moreadvanced analysis generally uses the vertical format.Consider this pattern:x‐valuey‐value0619212315418521To analyze the pattern, we calculate differences of successive values in the table. These arecalled first differences. If the first differences are constant, we can proceed to converting thepattern into an equation. If not, we do not have a linear pattern. In this case, we may chooseto continue by calculating differences of the first differences, which are called seconddifferences, and so on until we get a pattern we can work with.In the example above, we get a constant set of first differences, which tells us that the patternis indeed linear.x‐valuey‐valueFirst Differences061932123315341835213Converting a Linear Pattern to an EquationCreating an equation from the pattern is easy if you haveconstant differences and a y‐value for x 0. In this case, The equation takes the form 𝒚 𝒎𝒙 𝒃, where “m” is the constant difference from the table, and “b” is the y‐value when x 0.In the example above, this gives us the equation: 𝒚𝟑𝒙𝟔.Note: If the table does not have avalue for x 0, you can still obtainthe value of “b”. Simply extend thetable left or right until you have anx‐value of 0; then use the firstdifferences to calculate what thecorresponding y‐value would be.This becomes your value of “b”.Finally, it is a good idea to test your equation. For example, if 𝑥 4, the above equation gives𝑦3 46 18, which is the value in the table. So we can be pretty sure our equation iscorrect.Version 3.4Page 11 of 187April 6, 2022

BasicsADVANCEDChapter 1AlgebraIdentifying Number PatternsWhen looking at patterns in numbers, is is often useful to take differences of the numbers youare provided. If the first differences are not constant, take differences again.n‐3‐11357 n2510172637 2357911222222222When first differences are constant, the pattern represents alinear equation. In this case, the equation is: y 2x ‐ 5 . Theconstant difference is the coefficient of x in the equation.When second differences are constant, the pattern represents aquadratic equation. In this case, the equation is: y x 2 1 . Theconstant difference, divided by 2, gives the coefficient of x2 in theequation.When taking successive differences yields patterns that do not seem to level out, the patternmay be either exponential or recursive.n5711193567n23581321Version 3.4 2248163224816 2123581123In the pattern to the left, notice that the first and seconddifferences are the same. You might also notice that thesedifferences are successive powers of 2. This is typical for anexponential pattern. In this case, the equation is: y 2 x 3 .In the pattern to the left, notice that the first and seconddifferences appear to be repeating the original sequence. Whenthis happens, the sequence may be recursive. This means thateach new term is based on the terms before it. In this case, theequation is: y n y n‐1 y n‐2 , meaning that to get each new term,you add the two terms before it.Page 12 of 187April 6, 2022

BasicsADVANCEDChapter 1AlgebraCompleting Number PatternsThe first step in completing a number pattern is to identify it. Then, work from the right to the left, filling inthe highest order differences first and working backwards (left) to complete the table. Below are twoexamples.Example 1Example 2n‐162562123214n‐162562123214n‐162562123214 271937619112182430666 2 71937619112182430n ‐176192537626112391214127341169510217727 33666666 2 312182430364248666666Consider in the examples the sequences of sixnumbers which are provided to the student. You areasked to find the ninth term of each sequence.n23581321Step 1: Create a table of differences. Take successivedifferences until you get a column of constantdifferences (Example 1) or a column that appears torepeat a previous column of differences (Example 2).n23581321Step 2: In the last column of differences you created,continue the constant differences (Example 1) or therepeated differences (Example 2) down the table.Create as many entries as you will need to solve theproblem. For example, if you are given 6 terms andasked to find the 9th term, you will need 3 ( 9 ‐ 6)additional entries in the last column.n23581321Step 3: Work backwards (from right to left), filling ineach column by adding the differences in the columnto the right.n23581321345589In the example to the left, the calculations areperformed in the following order:2Column : 30 6 36; 36 6 42; 42 6 48Column : 91 36 127; 127 42 169; 169 48 217Column n: 214 127 341; 341 169 510; 510 217 727 2 3123581123011 2 123581123 2 312358132134112358130112353011235The final answers to the examples are the ninth items in each sequence, the items in bold red.Version 3.4Page 13 of 187April 6, 2022

Chapter 1BasicsAlgebraReal Number SetsNumber SetSymbolNatural (or,Counting) NumbersN orZ DefinitionExamplesNumbers that you wouldnormally count with.1, 2, 3, 4, 5, 6, Whole NumbersWAdd the number zero to theset of Natural Numbers0, 1, 2, 3, 4, 5, 6, IntegersZWhole numbers plus the setof negative Natural Numbers ‐3, ‐2, ‐1, 0, 1, 2, 3, Any number that can beRational NumbersQexpressed in the formAll integers, plus fractions andmixed numbers, such as:,2 174,, 3365where a and b are integersand 𝑏 0.Real NumbersRAll rational numbers plus rootsand some others, such as:Any number that can bewritten in decimal form,even if that form is infinite. 2 , 12 , π, eReal Number Set TreeReal ersion 3.4IrrationalFractions andMixed NumbersNegativeIntegersZeroPage 14 of 187April 6, 2022

Chapter 2OperationsAlgebraOperating with Real NumbersAbsolute ValueThe absolute value of something is the distance it is from zero. The easiest way to get theabsolute value of a number is to eliminate its sign. Absolute values are always positive or 0. 5 5 3 3 0 1.5 01.5Adding and Subtracting Real NumbersAdding Numbers with the Same Sign: Adding Numbers with Different Signs: Add the numbers without regardto sign.Give the answer the same sign asthe original numbers.Examples:63912 6 18 Ignore the signs and subtract thesmaller number from the larger one.Give the answer the sign of the numberwith the greater absolute value.Examples:633711 4Subtracting Numbers: Change the sign of the number or numbers being subtracted.Add the resulting numbers.Examples:6363313 4 1349Multiplying and Dividing Real NumbersNumbers with the Same Sign: Version 3.4Numbers with Different Signs: Multiply or divide the numberswithout regard to sign.Give the answer a “ ” sign.Examples:6 318 1812 34 4 Page 15 of 187Multiply or divide the numbers withoutregard to sign.Give the answer a “-” sign.Examples:6 3181234April 6, 2022

Chapter 2OperationsAlgebraProperties of AlgebraProperties of Addition and Multiplication. For any real numbers a, b, and c:PropertyDefinition for Addition𝑎Closure Property𝑏 is a real number𝑎Identity Property0𝑎Inverse Property𝑎0𝑎𝑎Commutative PropertyAssociative PropertyDefinition for Multiplication𝑏𝑎𝑎𝑎𝑏𝑏𝑐𝑎𝑎 1𝑎0𝐹𝑜𝑟 𝑎𝑎𝑏𝑐𝑎 𝑏Distributive Property𝑎 𝑏 is a real number𝑐1 𝑎𝑎0, 𝑎 1𝑎1 𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏 𝑐𝑎 𝑏1𝑎 𝑏 𝑐𝑎 𝑐Properties of Zero. For any real number a:Multiplication by 00 Divided by SomethingDivision by 0Version 3.4𝑎 00 𝑎𝐹𝑜𝑟 𝑎0,00is undefined even if aPage 16 of 1870April 6, 2022

Chapter 2OperationsAlgebraProperties of AlgebraOperational Properties of Equality. For any real numbers a, b, and c:PropertyDefinitionAddition Property𝐼𝑓 𝑎𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐𝑏𝑐Subtraction Property𝐼𝑓 𝑎𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐𝑏𝑐𝐼𝑓 𝑎Multiplication Property𝐼𝑓 𝑎Division Property𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑏 𝑎𝑛𝑑 𝑐𝑏 𝑐0, 𝑡ℎ𝑒𝑛 𝑎𝑐𝑏𝑐Other Properties of Equality. For any real numbers a, b, and c:PropertyDefinition𝑎Reflexive Property𝐼𝑓 𝑎Symmetric Property𝐼𝑓 𝑎Transitive PropertySubstitution PropertyVersion 3.4If 𝑎𝑎𝑏, 𝑡ℎ𝑒𝑛 𝑏𝑏 𝑎𝑛𝑑 𝑏𝑎𝑐, 𝑡ℎ𝑒𝑛 𝑎𝑐𝑏, then either can be substituted for theother in any equation (or inequality).Page 17 of 187April 6, 2022

Chapter 3Solving EquationsAlgebraSolving Multi‐Step EquationsReverse PEMDASOne systematic way to approach multi‐step equations is Reverse PEMDAS. PEMDAS describesthe order of operations used to evaluate an expression. Solving an equation is the opposite ofevaluating it, so reversing the PEMDAS order of operations seems appropriate.The guiding principles in the process are: Each step works toward isolating the variable for which you are trying to solve.Each step “un‐does” an operation in Reverse PEMDAS sInversesNote: Logarithms are theinverse operator to exponents.This topic is typically covered inthe second year of Algebra.Remove Parentheses (and repeat process)The list above shows inverse operation relationships. In order to undo an operation, youperform its inverse operation. For example, to undo addition, you subtract; to undo division,you multiply. Here are a couple of examples:Example 2Example 1144Solve:Step 1: Add 32 2𝑥53𝑥3183Result:Step 2: Divide by 22 2𝑥2522𝑥6Result:2𝑥Step 3: Remove parentheses51Result:Step 4: Subtract 52𝑥5515Result:2𝑥Solve:Step 1: Add 43𝑥Result:Step 2: Divide by 3Result:44Notice that we add and subtract before wemultiply and divide. Reverse PEMDAS.33536With this approach, you will be able toStep 5: Divide by 222solve almost any multi‐step equation. AsResult:𝑥3you get better at it, you will be able to usesome shortcuts to solve the problem faster.Since speed is important in mathematics, learning a few tips and tricks with regard to solvingequations is likely to be worth your time.Version 3.4Page 18 of 187April 6, 2022

Chapter 3Solving EquationsAlgebraTips and Tricks in Solving Multi‐Step EquationsFractional CoefficientsFractions present a stumbling block to many students in solving multi‐step equations. Whenstumbling blocks occur, it is a good time to develop a trick to help with the process. The trickshown below involves using the reciprocal of a fractional coefficient as a multiplier in thesolution process. (Remember that a coefficient is a number that is multiplied by a variable.)Example 1𝑥Solve:Explanation: Since is the reciprocal of ,8when we multiply them, we get 1, and 1 𝑥 𝑥. Using this approach, we can avoiddividing by a fraction, which is more difficult.Multiply by : Result:𝑥 812Example 2Explanation:𝑥Solve:Multiply byResult:4: 𝑥42 2 , sowhen we multiply them, we get 1. Noticethe use of parentheses around the negativenumber to make it clear we are multiplyingand not subtracting.444 is the reciprocal of8Another Approach to ParenthesesIn the Reverse PEMDAS method, parenthesesare handled after all other operations.Sometimes, it is easier to operate on theparentheses first. In this way, you may be ableto re‐state the problem in an easier form beforesolving it.Example 3, at right, is another look at theproblem in Example 2 on the previous page.Use whichever approach you find most to yourliking. They are both correct.Version 3.4Example 3Solve:2 2𝑥 53Step 1: Distribute the lead multiplier (2)5Result:4𝑥Step 2: Combine constants5103Result:Step 3: Subtract 74𝑥Result:Step 4: Divide by 44𝑥4124𝑥3Result:Page 19 of 1877757April 6, 2022

Chapter 4Probability & StatisticsAlgebraProbability and OddsProbabilityProbability is a measure of the likelihood that an event will occur. It depends on the number ofoutcomes that represent the event and the total number of possible outcomes. In equation terms,𝑷 𝒆𝒗𝒆𝒏𝒕𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒊𝒏𝒈 𝒕𝒉𝒆 𝒆𝒗𝒆𝒏𝒕𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒐𝒕𝒂𝒍 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔Example 1: The probability of a flipped coin landing as a head is 1/2. There are two equally likely eventswhen a coin is flipped – it will show a head or it will show a tail. So, there is one chance out of two thatthe coin will show a head when it lands.𝑃 ℎ𝑒𝑎𝑑1 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑓 𝑎 ℎ𝑒𝑎𝑑2 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠12Example 2: In a jar, there are 15 blue marbles, 10 red marbles and 7 green marbles. What is theprobability of selecting a red marble from the jar? In this example, there are 32 total marbles, 10 ofwhich are red, so there is a 10/32 (or, when reduced, 5/16) probability of selecting a red marble.10 𝑟𝑒𝑑 𝑚𝑎𝑟𝑏𝑙𝑒𝑠32 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑟𝑏𝑙𝑒𝑠𝑃 𝑟𝑒𝑑 𝑚𝑎𝑟𝑏𝑙𝑒1032516OddsOdds are similar to probability, except that we measure the number of chances that an event will occurrelative to the number of chances that the event will not occur.𝑶𝒅𝒅𝒔 𝒆𝒗𝒆𝒏𝒕𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒊𝒏𝒈 𝒕𝒉𝒆 𝒆𝒗𝒆𝒏𝒕𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔 𝑵𝑶𝑻 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒊𝒏𝒈 𝒕𝒉𝒆 𝒆𝒗𝒆𝒏𝒕In the above examples,𝑂𝑑𝑑𝑠 ℎ𝑒𝑎𝑑1 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑓 𝑎 ℎ𝑒𝑎𝑑1 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑓 𝑎 𝑡𝑎𝑖𝑙11𝑂𝑑𝑑𝑠 𝑟𝑒𝑑 𝑚𝑎𝑟𝑏𝑙𝑒10 𝑟𝑒𝑑 𝑚𝑎𝑟𝑏𝑙𝑒𝑠22 𝑜𝑡ℎ𝑒𝑟 𝑚𝑎𝑟𝑏𝑙𝑒𝑠1022 Note that the numerator and the denominator in an odds calculation add to the total number ofpossible outcomes in the denominator of the corresponding probability calculation. To the beginning student, the concept of odds is not as intuitive as the concept of probabilities;however, they are used extensively in some environments.Version 3.4Page 20 of 187511April 6, 2022

Chapter 4Probability & StatisticsAlgebraProbability with DiceSingle DieProbability with a single die is based on the number of chances of an event out of 6 possibleoutcomes on the die. For example:𝑃 2𝑃 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟𝑃 𝑛𝑢𝑚𝑏𝑒𝑟5Two DiceProbability with two dice is based on the number of chances of an event out of 36 possibleoutcomes on the dice. The following table of results when rolling 2 dice is helpful in this regard:1st Die2nd 9101112The probability of rolling a number with two dice is the number of times that number occurs inthe table, divided by 36. Here are the probabilities for all numbers 2 to 12.𝑃 2𝑃 5𝑃 8𝑃 11𝑃 3𝑃 6𝑃 9𝑃 12𝑃 4𝑃 7𝑃 10𝑃 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟𝑃 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 3𝑃 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟𝑃 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4𝑃 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 6Version 3.4Page 21 of 187April 6, 2022

Chapter 4Probability & StatisticsAlgebraCombinationsSingle Category CombinationsThe number of combinations of items selected from a set, several at a time, can be calculatedrelatively easily using the following technique:Technique: Create a ratio of two products. In the numerator, start with the number oftotal items in the set, and count down so the total number of items being multiplied isequal to the number of items being selected. In the denominator, start with thenumber of items being selected and count down to 1.Example: How manycombinations of 3 items canbe selected from a set of 8items? Answer:8 7 63 2 1Example: How manycombinations of 4 items canbe selected from a set of 13items? Answer:13 12 11 104 3 2 156715Example: How manycombinations of 2 items canbe selected from a set of 30items? Answer:30 292 1435Multiple Category CombinationsWhen calculating the number of combinations that can be created by selecting items fromseveral categories, the technique is simpler:Technique: Multiply the numbers of items in each category to get the total number ofpossible combinations.Example: How many differentpizzas could be created if youhave 3 kinds of dough, 4 kindsof cheese and 8 kinds oftoppings?Answer:3 4 8Version 3.496Example: How many differentoutfits can be created if youhave 5 pairs of pants, 8 shirtsand 4 jackets?Answer:5 8 4160Page 22 of 187Example: How many designsfor a car can be created if youcan choose from 12 exteriorcolors, 3 interior colors, 2interior fabrics and 5 types ofwheels? Answer:12 3 2 5360April 6, 2022

Chapter 4Probability & StatisticsAlgebraStatistical MeasuresStatistical measures help describe a set of data. A definition of a number of these is provided in the table below:ConceptDescriptionData tionExample 1Example 235, 35, 37, 38, 4515, 20, 20, 22, 25, 54AverageAdd the values anddivide the total by thenumber of valuesMiddleArrange the values fromlow to high and take themiddle value(1)3721(1)MostThe value that appearsmost often in the dataset3520SizeThe difference betweenthe highest and lowestvalues in the data set45 – 35 1054 – 15 39OddballsValues that look verydifferent from the othervalues in the data setnone543535375384538151822222554626Notes:(1) If there are an even number of values, the median is the average of the two middle values. In Example 2, the median is 21,which is the average of 20 and 22.(2) The question of what constitutes an outlier is not always clear. Although statisticians seek to minimize subjectivity in thedefinition of outliers, different analysts may choose different criteria for the same data set.Version 3.4Page 23 of 187April 6, 2022

Chapter 5FunctionsAlgebraIntroduction to FunctionsDefinitions A Relation is a relationship between variables, usually expressed as an equation.In a typical x‐y equation, the Domain of a relation is the set of x‐values for which y‐ values can be calculated. For example, in the relation 𝑦 𝑥 the domain is 𝑥 0because these are the values of x for which a square root can be taken.In a typical x‐y equation, the Range of a relation is the set of y‐values that result for all values of the domain. For example, in the relation 𝑦 𝑥 the range is 𝑦 0 becausethese are the values of y that resu

Algebra Handbook Table of Contents Page Description Chapter 10: Polynomials - Basic 60 Introduction to Polynomials 61 Adding and Subtracting Polynomials