~ ~ = ~, DIVISION Z ~~~ ~ ~== ,m - - - - The f ollowing sets are infinite; that is, there are n~ last numbers The three dots indicate continuing or never-endmg patterns Counting or natural numbers = {1, 2, 3, 4, , 78, 79, } Whole numbers = {O, 1,2,3,4, ,296, 297, } Integers = { ••• , -4, -3, -2, -1, 0, 1, 2, 3, } R ational num bers = {all numbers that can be written as fractions, p/q, where p and q are integers and q is not zero} RatlOna.1 numbers mclude all counting numbers, whole numbers and integers, 10 addlti.on to all proper and improper fraction numbers, and endmg or repeat 109 deCimal numbers Exs: 4/9, 3, -1 8.75, Jj6, - J25 Irrational num bers = {all numbers that cannot be expressed as rational num­ bers} As decimal numbers, irrational numbers not end nor repeat Exs: 3.171171117 , f"i, -.J2, 1t Real numbers = {all rational and all irrational numbers} OPERATIONS ABSOLUTE VALUE Absolute val ue is the distance (always positive) between a number and zero on the number line; the positive value ofa number Exs: 131 = 3; 1-31 = 3; 1-.51 =.5 ADDITION I Integers: When adding integers, follow these rules a If both numbers are positive, add them; the Sign of the answer WIll be posItive b lfboth numbers are negative, add them; the sign of the answer WIll be negative c Ifone number is negative and the other is positive (in any order), subtract the two numbers (even though they are joined by a plus sign); the sign ofthe answer will be the same sign as the sign of the number that has the larger absolute value Exs: + (-9) = -5; (-32) + (-2) = -34; (-12) + 14 = 2 Rational numbers: a.When adding two mixed number s, fractions, or decimal numbers, fol­ low the same sign rules that are used for integers (above), but also fol­ low the rules of operations for each type of number b.For mixed numbers and fractions, make sure the fractIOns have a com­ mon denominator, then add the numbers Mixed numbers and fractIOns can also be changed to decimal numbers and then added c.For decimal numbers, line the decimal points up, then add the numbers, bringing the decimal point straight down Exs: (-4 1/2) + (5 3/4) = (-4 2/4) + (5 3/4) = 1/4; 5.667 + (-.877) = 4.79 Irrational num bers: a When adding irrational numbers, exact decimal values cannot b~ used If decimal values are used, then they are rounded and the answer IS only an approximation Instead, if the two irrational numbers are multiples of the same sq uare root, radical expression, or p.i (n), then simply add the coefficients (numbers in front) of the roots or pi (n) Exs: 4.)3 +5.)3 =9 f 3;(-61t)+91t=31t ; 3f"i +3.J2 cannot be added any further because the two square roots are different SUBTRACTION I Subtraction of all categories of numbers can be accomplished by adding the opposite of the number to be subtra.cted After changing the sign ofthe number ill back of the mInUS Sign, follow the rules addition as stated above Exs: 8- (-3)=8+(+3)=11;(-15)-(9) =(-15)+(-9) = - 24 MULTIPLICATION Integers: When dividing integers, follow these ru.le ~: a If the signs of the numbers are the same, divide th em and make the answer positive b If the signs of the numbers are different, divide the m and make the answer negative c The sign ofthe answer does not come from the number with the larger absolute value as it does in addition Exs: (-30)/(5) = - 6; (-22)/(-2) = 11; (70)/(-10) = -7 Rational numbers: a When dividing rational numbers, follow the sign rules that are used for di­ viding integers (listed above) and the rules for dividing ea~h type of number b For mixed numbers, change each mixed number to an Improper fractIOn, invert or flip the number behind the division sign and follow the rules for multiplying fractions c For decimal numbers, first move the deCimal polOt 10 the diVisor to the back of the number, then move the decimal point the same nU'!lber of po­ sitions to the right in the dividend Divide the numbers, then brmg the dec­ imal point straight up into the quotient (answer) Additional zeros can be written after the last digit behind the decimal pomt 10 the dIvidend so the division process can continue if needed Irrational numbers: a When dividing irrational numbers, follow the same sign rules that are used for dividing integers (listed above) b Ifradical expressions are divided and they have the same mdlce ~ , ~hen the numbers (radicands) under the root symbols (radicals) can be divi ded Exs: (-M)/ (.)3)=-f5; (-J30 )/( v6 )=-f5; W/.J2 cannot be divided, only simplified as demonstrated in the Quick Study®Algebra Part One study guide EXPONENTS/POWERS I Definition: an =~ , that is, the number written in the upper right-hand n corner is called the exponent or power, and it tells how many times the other number (called the base) is mul tiplied times itself If an exponent cannot be seen, it equals l Exs: 56 = - 5.- ­ - - = 15,625; notice that the base, 5, was multiphed times Itself times because the exponent was Rule: an_ am = a m+ n ; that is, when multiplying the same base, the new exponent can be found quickly by adding the exponents of the bases that are multJphed Exs: (53) (54) = 57; (3 2) (7 3) (72) (3 S) = (3 7) (7 S) Rule: ant am= a n- m; that is, when dividing the same base, the new exponent can be found quickly by subtracting the exponents of the bases that are di­ vided The new base and exponent go either in the numerato r or 10 the de­ nominator, wherever the highest exponent was located 10 the ongmal prob­ lem Exs: (7S) / (7 2) = 73; (3 4) / (3 6) = / (3 2) Rule: a-I = lIa; and lIa-1 = a; that is, a negative exponent can be changcd to a positive exponent by moving the base to the other section of the fraction ; numerator goes to denominator or denominator goes to numerator Exs: 7-3 = 11(7 ); 11(5-2 ) = 52; 3(2-4) = 3/(24); notice the stayed in the nu­ merator because the invisible exponent is always positive l Rule: (aO)m = a om ; that is, when there is a base with an exponent raised to an­ other exponent, then the short cut is to multiply the exponents Ex: (-3Z 2)3 = (_3)3(Z2)3 = -27z ORDER OF OPERATIONS I Integers: When multiplying integers, follow these rules When a problem has many operations, the order in which the operations are a If the signs ofthe numbers are the same, mul~ply $1d make the answer poslJ;ive completed will give different answers; so, there is an order ofoperatio,!s rules b Ifthe signs ofthe numbers are different, mulnply and make the answer negative Do the operations in the parentheses (or any enclosure symbols) fIrst c NOfE: The sign.of the answer does not come from the number WIth the larger ab­ Do any exponents or powers next solute value as it does in addition Exs: (- 4)(5)= - 20;(-3)(-2) = 6; (7)(-10)= -70 Do any multiplication and division, go~g I~ft to nght 1I1 th~ or~er they appear Rational numbers: (this means division is done before multlphcatJon If It comes first 1I1 the problem) a When multiplying rational numbers, follow the sign rules that are used for Do the addition and subtraction, going left to right in the order they appear (thiS multiplying integers (above) and the rules for multiplying each type ofnum.ber means subtraction is done before addition ifit comes flTSt in the problem) b For mixed numbers, change each mixed number to an Improper fractIOn, Exs:4+2 (3+7)=4+2 (10)=4+20=24; 4075-2 +474=8-2+ = 16+ = 17 and then multiply the resulting fractions SCIENTIFIC NOTATION c For fractions, multiply the numerators and the denommators, then reduce A fo rm ofa decimal number where the decimal point i.~ always behind exthe answer actly one non-zero digit and the number is multiplied by a power often d For decimal numbers, multiply them as though they were integers, then put the Exs: 4.87 x 108 ; 3.981 X W - decimal point in the answer so there is the same number of di~ts behind the dec­ imal point in the answer as there are behind both decunal POillts ill the problem It is a method for representing very large or very small numbers :-vitho ut writing a lot of di gits Ex: 243,700,000,000,000 would be wntten as Irrational num bers: 2.437 x 10 14 ; 000000982 would be written as 9.82 x 10-7 a When multiplying irrational numbers, follow the same sign rules that are A positive or zero exponent on the 10 means the number value is more than or equal used for integers (listed above) to one A negative exponent on the 10 means the number value 1S less than one b If radical expressions are multiplied and they have the same mdlces? then the Ex: 5.29 x 10-10 = 000000000529 and 5.29 x 10 14 = 529,000,000,000,000 numbers (radicands) under the root symbols (radicals) can be multiplied Operations with very large or very small numbers can ~e completed using Exs: (-$)( f"i )=- J35; (3v1)(-4)=-12f"i the scientific notation form of the numbers, WIth calculators III , '" '" , m ALGEBRA CONCEPTS PROPERTIES DEFINITIONS I Add ition/Subtraction Property of Equality: If a = b, th en a + c = b + C and a - c = b - c; that is, you can add or subtract any number or term to or from an equation as long as you it on both sides of the equal sign Multip lication/Division Property of Equality: If a = b, then ac = bc and a/c = b/c (when c "# 0); that is, you can multiply or divide by any number or term as long as you it on both sides of the equal sign Remember, not divide by zero because it is undefined Symmetric Property: rfa = b, then b = a; that is, two sides of an equa­ tion can be exchanged without changing any signs or terms in the equation Ex: 3n + = S - 2n becomes S - 2n = 3n + A variable is a letter that represents a number A coefficient is a number that is multiplied by the variable It is found in front of a variable, but the multiplication sign is not written If the coeffi­ cient is one, the one is not written Exs: 5n = x n; if n were 3, then 5n would equalS x or 15 A term is a mathematical expression involving multiplication or division Terms are separated by an addition or subtraction sign Exs: 7a is one term; 3k + is two terms; 4m - Sm + is three terms SOLUTION METHODS Like (or similar) terms are terms that have the same variables and expo­ nents, written in any order The coefficients (numbers in front) not have to be the same Exs: 4m and 9m are like terms ; 5a 2c and -7a 2c are like terms; 3r3 and -9r2 are not Iike terms because the exponents are not the same; 15z4and St4 are not like terms because they not have the same variables FIRST DEGREE, ONE VARIABLE I Solving an equation means you are findin g the one numerical value that makes the equation true when it is put into the equation in place of the variable Using inverse operations is the best method for first-degree equations Using in­ verse operations means you the operation opposite to the one in the equation One-step eq uations: Equations having only one operation (+, -, x, or ;.) with the variable require only one inverse operation If the equation has ad­ dition, then you subtraction; if subtraction, you add ition; if multipli­ cation, you division; if division, you multiplication Ex 1: n + = -3 ~ is added to n, so, ~ Subtract from both sides n + - = -3 - n = -10 ~ giving the solution of-1O OPERATIONS & PROPERTIES I Addition and Subtraction: Only like (or similar) terms can be added or subtracted Once it is determined that the terms are like terms, only the co­ efficients (numbers in front of the terms) are added or subtracted Exs: 3n + 7n - 11n = IOn - 11n = IOn + (-11n) = -In or simply -n; Ex 2: j = ~ a is divi ded by 3, so, 1· = • ~ Multiply by on both sides 14k2 + 5n - 10k2 - n = 4k2 + 4n Multiplication: Any terms can be multiplied They not have to be like terms When multiplying terms, multiply the coefficients (numbers in front) and the matching variables Ex: (-3m2n)(5m4n) = -15m n 2; remember, when multiplying, make sure the bases are the same, then add exponents a = 27 ~ giving the solution of27 Two-step eq uatio ns: a Equations that have two operations connected to the variable require two operations that are the opposites of the ones that are in the equation It is much easier to addition or subtraction before doing multiplication or division This is the opposite of the order of operations because you are doing inverse or opposite operations to solve the equations Ex 1: 3x + = - S ~ was added, so, 3x + - = -S - ~ subtract on both sides 3x ; = -12 ; ~ was mUltiplied, so divide by on both sides x = - ~ giving the solution of - Division: Any terms can be divided They not have to be like terms Division is usually written in fraction form When dividing terms, divide or reduce the coefficients (numbers in front) and the matching variables Remember that, to divide with exponents, you must subtract the exponents once you match the same bases Ex: (30a7c2)/(-6a4c3d2) = (-5a 3)/(cd 2) be­ cause 30 divided by -6 is -5, a 7divided by a is a\ c3divided by c2 is c, and there is no other variable d to divide by the d 2, so it remains the same Ex 2: ~ - = ~ was subtracted, so, Commutative Property: a + b = b + a and a - b = a + (-b) = (-b) + a ; there­ fore, terms can be moved as long as you take the proper sign (negative or positive) with the term Exs: 4p2 + Sp3 = Sp3 + 4p2; 14c - 3f= (-3t) + 14c ~- 7+7= 3+7 ~ add to both sides !l • = 10 • ~ n is divided by 2, so multiply by on both sides Associative Property: (a + b) + c = a + (b + c) and (a - b) - c = a + (-b + -c); therefore, terms can be added in any order as long as all subtraction is first changed to addition Ex: (5j - Sj) - 12j = 5j + (-8j + -12j) n = 20 ~ giving the solution of 20 b If the equation has the variable on the right side of the equal sign, then it can be solved, leaving the variable on the ri ght side, or it can be turned Distributive Property: a(b + c) = ab + ac and a(b - c) = ab - ac; therefore, around by simply taking everything on each side of the equal sign and put­ if the terms inside the parentheses cannot be added or subtracted, multiply ting it on the opposite side without chang ing any signs or terms in any way them BOTH or ALL by the value located in front of the parentheses Exs: (symmetric property) 2 2 3 3n(5n + 6) = 15n + ISn; a c(5a + 2ac - c ) = 5a c + 2a c - a c More tha n two-step equations: Eq uations sometimes req uire sim plifyin g each side of the equation separately before beginning to inverse operations Double Negative Property: - (-a) = a; therefore, if there is a negative of Ex: 3(2n + 1) + = 4n - 10 ~ di stribu te a negative, it becomes positive, just like a negative number times a nega­ 6n + + = 4n - 10 ~ add like terms tive number equals a positive number 6n + 12 = 4n - 10 ~ now begin inverse operations 6n + 12 - 40 = 4n - 10 - 4n ~ subtract 4n on both sides TRANSLATING 2n + 12 - 12 = -10 - 12 ~ subtract 12 on both sides 2n ; = -22 ; ~ divide by on both sides PUTTING WORDS INTO ALGEBRAIC STATEMENTS ~ giving the sol ution of -II n = -11 There are several key words or phrases that often help in Proportions: Equations in which both sides of the equa l sign are fractions converting words into algebraic statements The cross-multiplication rule can be used to solve such equations I Addition: Plus; add; more than; increased by; sum; total Exs: "4 more than The rule is that if !! = !l , then ad = bc a number" becomes + n; "a number increased by 3" becomes n + e d Subtraction: Minus; subtmct; decreased by; less than; difference Exs: "6 less Ex 1:1 =1 than a number" becomes n - It cannot be written - n, because the is being x taken away from the number, not the other way around; "a number decreased by • = • x ~ cross multiply 5" becomes n - and not - n; always consider which value is being subtmcted 35 ; = 3x ; ~ inverse operation, divide by 3 Multiplication: Times; multiply; product; of (when used with a fraction); 11 = x ~ solution doubled; tripled Exs: " 2h of a number" becomes (2/3)n; "the product of and a number" becomes 7n Ex 2:.! = _ 3_ (x +2) Division: Divided by; divided into; quotient; a half (divide by 2); a third (divide by 3) Exs: "A number divided by 2" becomes nl2; "the quotient ofS and a number" 4(x + 2) = • ~ cross multiply becomes SIn 4x + S = 15 ~ distribute the Inequality and equality symbols: 4x + S - S = 15 - S ~ inverse operation; - S a > comes from "is greater than" or "is more than" and not "more than," ~ inverse operation; divi de by 4x ; = ; x = 1.75 ~ solution which is addition b < comes from " is less than" and not " less than," which is subtraction Grap hing solutions: Since equations have only one solution, the graphs of their c :::: comes from "is more than or equal to" or " is greater than or equal to." solutions are simply a solid dot on the number on the real number line Ex: If you d ::; comes from "is less than or equal to." solved the equation 4k-7 = -15 and found the answer e "# comes from "is not equal to." k = -2, then you would draw a real number line and "_~ I -\ f = comes from "is equal to" or " equals." put a solid dot on the line above -2, such as at right t f GRAPHING LINES There are many ways to graph a linear equation ALGEBRAIC INEQUALITIES Algebraic inequalities are statements that not have an equal sign but rather one of these symbols: >, < , ~ , ~ , or 7= PROPERTIES I Pick any number to be the value of the x variable Put it into the equation for the x and then solve the equation for the y This gives one ordered pair (x,y), with the number you picked followed by the nwnber you found when you solved the equation You should pick at least three different values for x and solve, giving points on the x x + 2y = Y line If the points don't t +- ,, " -+-' form a line, a mistake has 0 + 2y = y=2 been made on at least one of +- ,,-"' :, :-+-, the equation solutions I + 2y - 1.5 y =1.5 Ex: The linear equation I + -'~ +_, x + 2y = can be put into 2 + 2y = y=l a chart like the one at right: _ ;:._; .1._ Find the points where the line crosses the x-axis (called the X-intercept) and the y-axis (called the y-intercept) a This can be done by putting a zero into the equation for the x variable and solving for the y This gives the point where thc line crosses the y-axis because all points on the y-axis have x numbers of zero b Next, put a zero into the equation for the y number and solve for the x This gives the point _-, ,.- where the line crosses x 3x - y = y y the x-axis because all t + -+ f (1%,0) 3' - y = -5 P oints on the x-axis Y=-5 have y numbers of zero t-1""'1-+ ' + f Ex: The linear equation 3x - = (0, 5) 3x - Y= could be put into x = ,1 a chart like the one at right: - - - - ' - .- c Find one point on the line by: (I) Putting a number into the equation for the x and solving for the y (2) Next, use the slope of the line The slope can be found in the equation Look at the coefficient (nwnber in front) of the x variable, change the sign of this number and divide it by the coefficient of the y variable This is the slope of the line (3) Then, graph the point you found and count the slope from that point using (rise)/(run) Ex: The linear equation 2x - Y = goes through the point (3, -1) (3,-1) The slope is -2/ -] because you change the sign of rise =-2 the number in front ofthe x variable and divide it by run =- the coefficient of the y variable, which is -] Graph these values as at right I Addition/Subtraction Property of Inequality: If a > b, then a + c > b + c and a - c > b - c Also, if a < b, then a + c < b + c and a - c < b - c This means that you can add or subtract any number or term to or from both sides of the inequality MultiplicationlDivision Property of Inequality: If a> b, then ac > bc and a/c > b/c only if c is a positive number If c is a negative number, then a > b becomes ac < bc or a/c < b/c (Notice that if > and you multiply each side by -2, then you get -16 > -10, which is false, but if you tum the symbol around, getting -16 < -10, it becomes true again.) Caution: Tum the symbol around only when you multiply or divide by a negative number SOLUTION METHODS FIRST DEGREE, ONE VARIABLE Solving inequalities is exactly the same as solving equations, as discussed on page 2, with only one exception The exception is when you multiply or di­ vide by a negative number, the inequality symbol turns around to keep the inequality true, so you will get a true solution The symbol does not tum around when you are adding or subtracting any terms or numbers or when you are multiplying or dividing by a positive number Ex: 3(x + 2) > -15 3x + > -15 f- distribute the 3x + - > -15 - f- inverse operation (- 6) f- inverse operation (-+- 3) 3x -+- > -21 -+- x> -7 f- solution Note: The > symbol did NOT turn around because the division was by + 3, not - 21 Graphing solutions: Inequalities have many solutions or answers, so the graphs of the solutions look very different from the graphs of equations a Graphs of equations usually have only one solid dot, but the graphs of in­ equalities have either solid dots with rays or open dots with rays Ex: If the solution to an inequality is x > -7, the graph at right is with an •I E9 I •• open dot because -7 does NOT make the in-8 -7 -6 -5 equality true, only numbers more than -7 b The solid dot shows that the number is part of the answer, but an open dot shows that the number is not part of the answer but only a beginning point Ex: If the solution is n ~ 3, the graph at • I I • I I '" right shows a solid dot I It A - COORDINATE PLANE GRAPHING INEQUALITIES On the coordinate plane, linear inequalities are lille graphs with a shaded region included, either above or below the line POINTS I The coordinate plane is a grid with an x-axis and a y-axis Every point on a plane can be named using an ordered pair An ordered pair is two numbers separated by a comma and enclosed by parentheses (x,y) The first number is the x nwnber and the second number is the y number Ex: (3, -5), where x = and y = -5 The point where the x-axis and the y-axis intersect or -+ cross is called the origin and has the ordered pair (0,0) The x number in the ordered pair tells you how far to go to the right (if positive) or to the left (if negative) from the origin (0,0) The y number in the ordered pair tells you how far to go up (if positive) or down (if negative), either from the ori­ gin or from the last location found by using the x nwnber Graph the line (even if the inequality does not include the equal sign, you must graph the corresponding equality) Pick a point above the line Put the number values for x and for y into the inequality to see if they make the inequality true If the point makes the inequality true, shade that side of the line Tfthe point makes the inequality false, shade the other side of the line The actual line is drawn as a solid line if the inequality includes the equal sign The actual line is drawn as a dashed line if the inequality does not include the equal sign Ex: Graph x + y < x LINES & EQUATIONS I The coordinate plane and ordered pairs are used to name all ofthe points on a plane When the points form a line, a special equation can be written to represent all of the points on the line Since points are named using ordered pairs with x numbers and y numbers in them, equations of lines, called linear equations, are written with the vari­ ables x and/or y in them Exs: 2x + Y = 5; Y = x - 6; x = -2; Y = Lines that cross both the x-axis and the y-axis have equations that contain both the variables x and y Lines that cross the x-axis and not cross the y-axis have equations that con­ tain only the variable x and not the variable y Lines that cross the y-axis and not cross the x-axis have equations that con­ tain only the variable y and not the variable x slope = Yt -Yl X,-Xl = 0-(-2) =~=_1 -2-1 -3 or slope = rise run = x+0=2 x=2 r Test (0,0) in x+y ... This QUICKSTUDY'" guide outlines the major topics taught in Pre- Algebra courses For further deta il, see Algebra Part­ I and Algebra Part-2 Due to its condensed format, however, use it as a Pre- Algebra. .. -2, such as at right t f GRAPHING LINES There are many ways to graph a linear equation ALGEBRAIC INEQUALITIES Algebraic inequalities are statements that not have an equal sign but rather one of... not written Exs: 5n = x n; if n were 3, then 5n would equalS x or 15 A term is a mathematical expression involving multiplication or division Terms are separated by an addition or subtraction