Developmental Mathematics Second Edition

GeometryP Perimeter, A Area, C Circumference, V VolumePerimeter and AreaRectangleP 2l 2wSquareP 4sA lwA s2wTriangleP a b c1A bh2asParallelogramP 2a 2bA bhchhaahbblTrapezoidP a b c d1A h (b c )2cCircleC 2pr pdA pr 2rddbVolumeRectangular SolidRectangular Pyramid1V lwh3V lwhRight Circular Cone1V pr 2 h3Right Circular CylinderSphere4V pr 33V pr 2hrhhwhwlrhrlAngles Classified by MeasureAcuteRightObtuseStraight0 m A 90 m A 90 90 m A 180 m A 180 AAAATriangles Classified by SidesScaleneNo two sides are equal.IsoscelesAt least two sides are equal.RCBAPEquilateralAll three sides are equal.ZXQYTriangles Classified by AnglesAcuteAll three angles are acute.CARightOne angle is a right angle.ZRBPObtuseOne angle is obtuse.QXY

US Customary Systemof MeasurementLengthMetric Systemof MeasurementLength12 inches (in.) 1 foot (ft)3 feet 1 yard (yd)36 inches 1 yard5280 feet 1 mile (mi)Capacity1 millimeter (mm) 0.001 meter1 m 1000 mm1 centimeter (cm) 0.01 meter1 m 100 cm1 decimeter (dm) 0.1 meter1 meter (m) 1.0 meter1 m 10 dm1 dekameter (dam) 10 meters1 cup (c) 8 fluid ounces (fl oz)2 pints 1 quart (qt)2 cups 1 pint (pt) 16 fluid ounces4 quarts 1 gallon (gal)Weight16 ounces (oz) 1 pound (lb)2000 pounds 1 ton (T)Time60 seconds (sec) 1 minute (min)60 minutes 1 hour (hr)24 hours 1 day7 days 1 weekTemperatureCelsius (C) to Fahrenheit (F)9F C 325Fahrenheit (F) to Celsius (C)C 5 ( F - 32 )1 hectometer (hm) 100 meters1 kilometer (km) 1000 metersCapacity (Liquid Volume)1 milliliter (mL) 0.001 liter1 L 1000 mL1 liter (L) 1.0 liter1 hectoliter (hL) 100 liters1 kiloliter (kL) 1000 liters1 kL 10 hL1 milligram (mg) 0.001 gram1 g 1000 mg1 centigram (cg) 0.01 gram1 decigram (dg) 0.1 gram1 gram (g) 1.0 gram1 dekagram (dag) 10 grams1 hectogram (hg) 100 grams1 kilogram (kg) 1000 grams1g1 metric ton (t) 1000 kilograms1 kg 0.001 tWeight 0.001 kg1t 1000 kg 1,000,000 g 1,000,000,000 mg9US Customary and Metric EquivalentsLength1 in. 2.54 cm ( exact )1 ft 0.305 m1 yd 0.914 m1 mi 1.61 kmVolume1 cm 0.394 in.1 m 3.28 ft1 m 1.09 yd1 km 0.62 miArea1 in. 3 16.387 cm 31 ft 3 0.028 m 31 qt 0.946 L1 gal 3.785 L1 cm 3 0.06 in. 31 m 3 35.315 ft 31 L 1.06 qt1 L 0.264 galMass1 in. 2 6.45 cm 21 cm 2 0.155 in. 21 ft 2 0.093 m 21 m 2 10.764 ft 21 yd 2 0.836 m 21 acre 0.405 ha1 m 2 1.196 yd 21 ha 2.47 acres1 oz 28.35 g1 lb 0.454 kg1 g 0.035 oz1 kg 2.205 lb

Notation and TerminologyExponentsEquality and Inequality Symbolsa a a a . a ann factorsexponentbaseFractionsnumeratordenominatorab “is equal to” “is not equal to” “is less than” “is greater than” “is less than or equal to” “is greater than or equal to”Least Common Multiple (LCM)SetsGiven a set of whole numbers, the smallest number thatis a multiple of each of these whole numbers.The empty set or null set (symbolized or {   } ): A setwith no elements.RatiosThe union of two (or more) sets (symbolized ): The setof all elements that belong to either one set or the otherset or to both sets.abor a : bor a to b  A comparison of twoquantities by division.Proportionsa c b dA statement that two ratios are equal.The intersection of two (or more) sets (symbolized ):The set of all elements that belong to both sets.The word or is used to indicate union and the word andis used to indicate intersection.Algebraic and Interval Notation for IntervalsGreatest Common Factor (GCF)Given a set of integers, the largest integer that is a factor(or divisor) of all of the integers.Types of NumbersType of IntervalAlgebraicNotationIntervalNotationOpen Intervala x b(a, b)Closed Intervala x b a, b a x b a x b a, b)(a, b x a x b(a, )( - , b)a x a x b a, )( - , b aNatural Numbers (Counting Numbers):Half-openIntervalN { 1, 2, 3, 4, 5, 6, . }Whole Numbers: W { 0, 1, 2, 3, 4, 5, 6, . }Open IntervalIntegers: Z { ., 4, 3, 2, 1, 0, 1, 2, 3, 4, . }Rational Numbers: A number that can be written in theaformwhere a and b are integers and b 0.bIrrational Numbers: A number that can be written asan infinite nonrepeating decimal.Real Numbers: All rational and irrational numbers.Complex Numbers: All real numbers and the evenroots of negative numbers. The standard form of a complex number is a bi, where a and b are real numbers, ais called the real part and b is called the imaginary part.Absolute ValueaThe distance a real number a is from 0.Half-openIntervalGraphababababbbRadicalsThe symbolis called a radical sign.The number under the radical sign is called the radicand.The complete expression, such asradical or radical expression.In a cube root expression364, is called aa , the number 3 is called theindex. In a square root expression such asis understood to be 2 and is not written.The Imaginary Number ii 2-1 and i ( -1 )2 -1a, the index

Formulas and TheoremsPercentPA 100 BThe Pythagorean TheoremIn a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of(the percent proportion),the two legs: c 2 a 2 b 2whereP)100B base (number we are finding the percent of)A amount (a part of the base)cP% percent (written as the ratioR · B A (the basic percent equation),whereR rate or percent (as a decimal or fraction)B base (number we are finding the percent of)A amount (a part of the base)90 baProbability of an Eventprobability of an eventnumber of outcomes in event number of outcomes in sample spaceDistance-Rate-TimeProfitProfit: The difference between selling price and cost.The distance traveled d equals the productof the rate of speed r and the time t.Special Productsprofit selling price - cost1. x 2 - a 2 ( x a) ( x - a) :Percent of Profit:profitcost1.Percent of profit based on cost:2.Percent of profit based on selling price:Difference of two squares2. x 2 2ax a 2 ( x a) : Square of a binomial sum2profitselling price3. x 2 - 2ax a 2 ( x - a) : Square of a binomial difference2( ( x - a) ( x) ax a ) :4. x 3 a 3 ( x a) x 2 - ax a 2 : Sum of two cubesInterestSimple Interest: I P r tr Compound Interest: A P 1 n d rt5. x 3 - a 3ntContinuously Compounded Interest: A Pe rtwhereI interest (earned or paid)22Difference of two cubesChange-of-Base Formula for LogarithmsFor a, b, x, 0 and a, b 1, log b x log a x.log a bDistance Between Two PointsA amount accumulatedThe distance d between points P ( x1 , y1 ) and Q ( x 2 , y2 )P principal (the amount invested or borrowed)is d r annual interest rate in decimal or fraction formMidpoint Formulat time (one year or fraction of a year)n the number of times per year interest iscompoundede 2.718281828459 . . .( x 2 - x1 )2 ( y2 - y1 )2 .The midpoint between points P ( x1 , y1 ) and Q ( x 2 , y2 ) x x 2 y1 y2 is 1,. 22

Principles and PropertiesProperties of Addition and MultiplicationProperties of Rational Numbers (or ivePropertya b b aab baAssociativeProperty( a b) c a ( b c )a ( bc ) ( ab) cIdentitya 0 0 a aa 1 1 a aa ( - a) 01a 1 (a 0)aInverseZero Factor Law: a · 0 0 · a 0Pis a rational expression and P, Q, R, and K areQpolynomials where Q, R, S 0, thenThe Fundamental PrincipleP P K Q Q KMultiplicationP R P R Q S Q SDivisionP R P S Q S Q RAdditionP R P R Q QQSubtractionP R P-R- Q QQDistributive Property: a ( b c ) a b a cAddition (or Subtraction) Principle of EqualityA B, A C B C, and A - C B - C have thesame solutions (where A, B, and C are algebraicexpressions).Multiplication (or Division) Principle of EqualityA B A B, AC BC, andhave the same solutionsC C(where A and B are algebraic expressions and C is anynonzero constant, C 0).Properties of ExponentsProperties of RadicalsIf a and b are positive real numbers, n is a positive integer,m is any integer, andna is a real number thenm1.nab n a n b2.nab3.na anFor nonzero real numbers a and b and integers m and n:n n( )1mama n ( n a ) n ambm1The exponent 1a aThe exponent 0a0 1Properties of LogarithmsThe product rulea m a n a m nFor b 0, b 1, x, y 0, and any real number r,The quotient ruleam am - nan1.log b 1 03.x b log b x2.log b b 14.log b b x x5.log b xy log b x log b y The product rule6.log b7.log b x r r log b x1Negative exponentsa-n 1anPower rule(a ) aPower of a product(ab) n a n b nPower of a quotientan a nbbm nmnnZero Factor PropertyIf a and b are real numbers, and a · b 0, then a 0 orb 0 or both.1 a n a n (a m ) nor, in radical notation,4.x log b x - log b y The quotient ruleyProperties of Equations with Exponentsand LogarithmsFor b 0, b 1,1.If bx by, then x y.2.If x y, then bx by.3.If logbx logby, then x y (x 0 and y 0).4.If x y, then logbx logby (x 0 and y 0).

Equations and InequalitiesLinear Equation in x(First-Degree Equation in x)Absolute Value Inequalitiesax b c, where a, b, and c are real numbers and a 0.1. If x c, then - c x c.Types of Equations and their Solutions2. If ax b c, then - c ax b c.Conditional: Finite Number of Solutions3. If x c, then x -c or x c.Identity: Infinite Number of Solutions4. If ax b c, then ax b -c or ax b c.For c 0:Contradiction: No Solution(These statements hold true for and as well.)Linear InequalitiesQuadratic EquationLinear Inequalities have the following forms where a, b,and c are real numbers and a 0:ax b candax b cAn equation that can be written in the formax2 bx c 0, where a, b, and c are real numbers anda 0.ax b candax b cQuadratic FormulaCompound InequalitiesThe solutions of the general quadratic equationThe inequalities c ax b d and c ax b d arecalled compound linear inequalities.ax 2 bx c 0, where a 0, are x -b b 2 - 4ac.2aThe Discriminant(This includes c ax b d and c ax b d as well.)The expression b2 4ac, the part of the quadraticformula that lies under the radical sign, is called thediscriminant.Absolute Value EquationsFor statements 1 and 2, c 0:If b2 4ac 0, there are two real solutions.1. If x c, then x c or x c.If b2 4ac 0, there is one real solution, x -2. If ax b c, then ax b c or ax b c.b.2aIf b2 4ac 0, there are two nonreal solutions.3. If a b , then either a b or a b.4. If ax b cx d , then either ax b cx d orax b - ( cx d ) .Systems of Linear EquationsSystems of Linear Equations (Two Variables)The system is.consistent, andthe equations are independent.(One solution)inconsistent, andthe equations are independent.(No solution)consistent, andthe equations are dependent.(Infinite number of solutions)yyyxxx