Transcript Chapter 1.3
Intermediate Algebra – 1.3 •Operations with Real Numbers Three people were at work on a construction site. All were doing the same job, but when each was asked what the job was, the answers varied. “Breaking rocks,” the first replied. “Earning my living,” the second said.”Helping to build a cathedral,” said the third.” – Peter Schultz, German businessman Procedure - Addition • Adding numbers with the same sign • To add two numbers that have the same sign, add their absolute values and keep the same sign Procedure - Addition • Adding numbers with different signs • To add two numbers that have different signs, subtract their absolute values and keep the sign of the number with the greater absolute value. Procedure - Subtraction • For any real number a • a – b = a + (-b) Distance on number line • The distance between two points a and b is • d = |a – b| = |b – a| Procedure - Multiplying • When multiplying two real numbers that have different signs, the product is negative Procedure - Multiplying • When multiplying two numbers that have the same sign, the product is positive Procedure - multiplying • The product of an even number of negative factors is positive, • The product of an odd number of negative factors is negative. Division • Division by Zero is undefined. • 4/0 is undefined • 0/4 = 0 n a Procedure - Division a a a b b b Definition Square Root • For all real numbers a and b, if b a 2 then b is a square root of a Def: radicand • The number or expression under the radical symbol 2 x3 Def: Index of radical • The index is n n a 3 x b Calculator Keys • [+], [*], [/], [-], [^] • [ENTER] [2ND][ENTRY] ND • [2 ] [QUIT] [x,t,n] • [MODE] • [MATH][NUM][1:abs( ] Norman Vincent Peale: • “What seems impossible one minute becomes, …, possible the next. Section 1.4 • Intermediate Algebra • Properties of Real numbers (9) Commutative for Addition • a+b=b+a • 2+3=3+2 Commutative for Multiplication • ab = ba •2x3=3x3 •2*3=3*2 Associative for Addition • a + (b + c) = (a + b) + c • 2 + (3 + 4) = (2 + 3) + 4 Associative for Multiplication • (ab)c = a(bc) • (2 x 3) x 4 = 2 x (3 x 4) Distributive multiplication over addition • a(b + c) = ab + ac • 2(3 + 4) = 2 x 3 + 2 x 4 • X(Y + Z) = XY +XZ Additive Identity •a+0=a •3+0=3 •X+0=X Multiplicative Identity •ax1=a •5x1=5 •1x5=5 •Y*1=Y Additive Inverse • a(1/a) = 1 where a not equal to 0 • 3(1/3) = 1 George Simmel - Sociologist • “He is educated who knows how to find out what he doesn’t know.” Section 1.4 Intermediate Algebra • Apply order of operations • Please Excuse My Dear Aunt Sally. • P – E – M – D – A- S The order of operations • Perform within grouping symbols – work innermost group first and then outward. • Evaluate exponents and roots. • Perform multiplication and division left to right. • Perform addition and subtraction left to right. Grouping Symbols • • • • • • Parentheses Brackets Braces Radical symbols Fraction symbols – fraction bar Absolute value Algebraic Expression • Any combination of numbers, variables, grouping symbols, and operation symbols. • To evaluate an algebraic expression, replace each variable with a specific value and then perform all indicated operations. Evaluate Expression by Calculator • • • • • Plug in Use store feature Use Alpha key for formulas Table Program - evaluate The Pythagorean Theorem • In a right triangle, the sum of the square of the legs is equal to the square of the hypotenuse. a b c 2 2 2 Equation • A statement that two expression have the same value Intermediate Algebra – 1.5 • Walt Whitman – American Poet • “Seeing, hearing, and feeling are miracles, and each part and tag of me is a miracle.” 1.5 – Simplifying Expressions • Term – An expression that is separated by addition • Numerical coefficient – the numerical factor in a term • Like Terms – Variable terms that have the same variable(s) raised to the same exponential value Combining Like Terms • To combine like terms, add or subtract the coefficients and keep the variables and their exponents the same. example 7 3 4 x 2 7 3 4 x 2 11 3x H. Jackson Brown Jr. Author •“Let your performance do the thinking.” Integer Exponents • For any real number b and any natural number n, the nth power of b o if found by multiplying b as a factor n times. b bbb n N times b Exponential Expression – an expression that involves exponents • Base – the number being multiplied • Exponent – the number of factors of the base. Calculator Key • Exponent Key ^ Sydney Harris: • “When I hear somebody sigh,’Life is hard”, I am always tempted to ask, “Compared to what?” Intermediate Algebra 1.5 •Introduction •To •Linear Equations Def: Equation • An equation is a statement that two algebraic expressions have the same value. Def: Solution • Solution: A replacement for the variable that makes the equation true. • Root of the equation • Satisfies the Equation • Zero of the equation Def: Solution Set • A set containing all the solutions for the given equation. • Could have one, two, or many elements. • Could be the empty set • Could be all Real numbers Def: Linear Equation in One Variable • An equation that can be written in the form ax + b = c where a,b,c are real numbers and a is not equal to zero Linear function • A function of form • f(x) = ax + b where a and b are real numbers and a is not equal to zero. Def: Identity • An equation is an identity if every permissible replacement for the variable is a solution. • The graphs of left and right sides coincide. • The solution set is R R Def: Inconsistent equation • An equation with no solution is an inconsistent equation. • Also called a contradiction. • The graphs of left and right sides never intersect. • The solution set is the empty set. Def: Equivalent Equations • Equivalent equations are equations that have exactly the same solutions sets. • Examples: • 5 – 3x = 17 • -3x= 12 • x = -4 Addition Property of Equality • If a = b, then a + c = b + c • For all real numbers a,b, and c. • Equals plus equals are equal. Multiplication Property of Equality • If a = b, then ac = bc is true • For all real numbers a,b, and c where c is not equal to 0. • Equals times equals are equal. Solving Linear Equations • Simplify both sides of the equation as needed. – Distribute to Clear parentheses – Clear fractions by multiplying by the LCD – Clear decimals by multiplying by a power of 10 determined by the decimal number with the most places – Combine like terms Solving Linear Equations Cont: • Use the addition property so that all variable terms are on one side of the equation and all constants are on the other side. • Combine like terms. • Use the multiplication property to isolate the variable • Verify the solution Ralph Waldo Emerson – American essayist, poet, and philosopher (1803-1882) • “The world looks like a multiplication table or a mathematical equation, which, turn it how you will, balances itself.” Problem Solving 1.6 • 1. Understand the Problem • 2. Devise a Plan – Use Definition statements • 3. Carry out a Plan • 4. Look Back – Check units Types of Problems • • • • Number Problems Angles of a Triangle Rectangles Things of Value Les Brown • “If you view all the things that happen to you, both good and bad, as opportunities, then you operate out of a higher level of consciousness.” Types of Problems Cont. • Percentages • Interest • Mixture • Liquid Solutions • Distance, Rate, and Time Albert Einstein • “In the middle of difficulty lies opportunity.” Ralph Waldo Emerson – American essayist, poet, and philosopher (1803-1882) • “The world looks like a multiplication table or a mathematical equation, which, turn it how you will, balances itself.” Section 1.8 • Solve Formulas • Isolate a particular variable in a formula • Treat all other variables like constants • Isolate the desired variable using the outline for solving equations. Know Formulas • Area of a rectangle A = LW • Perimeter of a rectangle • P = 2L + 2W Formulas continued • Area of a square A s 2 • Perimeter of a square P 4s Formulas continued • Area of Parallelogram •A = bh Formulas continued • Trapezoid 1 A b1 b2 h 2 Formulas continued • Area of Circle A r • Circumference of Circle 2 C 2 r C d Formulas continued: • Area of Triangle 1 A bh 2 Formulas continued • Sum of measures of a triangle 081 3m 2m 1m o Formulas continued • Perimeter of a Triangle P s1 s2 s3 Formulas continued • Pythagorean Theorem a b c 2 2 2 Formulas continued: • Volume of a Cube – all sides are equal V s 3 Formulas continued • Rectangular solid V lwh • Area of Base x height Formulas continued • Volume Right Circular Cylinder V r h 2 Formulas continued: • Surface are of right circular cylinder S 2 rh 2 r 2 Formulas continued: • Volume of Right Circular Cone • V=(1/3) area base x height 1 2 V r h 3 Formulas continued: • Volume Sphere 4 3 V r 3 Formulas continued: • • • • • General Formula surface area right solid SA = 2(area base) + Lateral surface area SA=2(area base) + LSA Lateral Surface Area = LSA = (perimeter)*(height) Formulas continued: • Distance, rate and Time d = rt Interest I = PRT Useful Calculator Programs • • • • • • • • CIRCLE CIRCUM CONE CYLINDER PRISM PYRAMID TRAPEZOI APPS-AreaForm Robert Schuller – religious leader • “Spectacular achievement is always preceded by spectacular preparation.”