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Sect 1.1 Algebraic Expressions Variable Consist of variables and/or numbers, often with operation signs and grouping symbols. Any symbol that represents a number…letters or Constant A value that never changes. Variable Expression An expression that contains a variable. Evaluating the Expression To evaluate an expression, we substitute a value in for each variable in the expression and calculate the result. Area formula Perimeter A = (base)(height) = bh A = (length)(width) = lw P = all exterior sides added together. Rectangle: P = 2l + 2w Sect 1.1 added to sum of plus more than increased by 5 pounds was added to the number The sum of a number and 12 7 plus some number 20 more than the number The number increased by 3 n+5 x + 12 7+m r + 20 y+3 subtracted from difference of minus less than decreased by 2 was subtracted from the number difference of two numbers 8 minus some number 9 less than the number The number decreased by 10 w-2 a-b 8-c d-9 f - 10 Sect 1.1 multiplied by product of times twice of the number multiplied by 4 the product of two numbers 13 times some number twice the number half of the number divided by quotient of divided into ratio of per 3 divided by the number the quotient of two numbers 8 divided into some number the ratio of 9 to some number There were 28 miles per g gallons 4n xy 13z 2t 1 x 2 3q mn h 8 9 r 28 g Sect 1.1 Four less than Joe’s height in inches. h–4 Eighteen increased by a number. 18 + n A day’s pay divided by eight hours. p/ 8 Half of the pallet. 1/ p 2 Seven more than twice a number. 2x + 7 Six less than the product of two numbers. ab – 6 Nine times the difference of a number and 3. 9(m – 3) Eighty five percent of the enrollment. 85%(e) = 0.85(e) Twice the sum of a number and 3. 2(x + 3) The sum of twice a number and 3. 2x + 3 Sect 1.1 The symbol = (“equals”) indicates that the expressions on either side of the equal sign represents the same number. An equation is when two algebraic expression are equal to each other. Equations can be true or false. 4 8 32 True 32 = 32 94 6 False 5 = 6 x 5 13 We don’t know the value of x. In the last example, replacing the “x” with a value that makes the equation true is called a solution. Some equations have more than one solution, and some have no solutions. When all solutions have been found, we have solved the equation. Determine whether 8 is a solution of x + 5 = 13. 8 + 5 = 13 13 = 13 True, 8 is a solution Sect 1.1 When translating phrases into expressions to equations, we need to look for the phrases “is the same as”, “equal”, “is”, and “are” for the = sign. Translate. What number plus 478 is 1019? x + 478 = 1019 Twice the difference of a number and 4 is 24. 2 ( ________ x – _________ 4 ) = 24 “than” makes the terms switch around the minus sign Three times a number plus seven is the same as the number less than one. 3x + 7 = x – 1 The Taipei Financial Center, or Taipei 101, in Taiwan is the world’s tallest building. At 1666 ft, it is 183 ft taller than the Petronas Twin Towers in Kuala Lumpur. How tall are the Petronas Twin Towers? 1666 – 183 = P 1483 = P + 183 – 183 Sect 1.2 Equivalent expressions. 4 + 4 + 4, 3 4 , and 3(4) Laws that keeps expressions equivalent. Commutative Law for Addition switch around the times sign for Multiplication ab ba a b b a a b ba switch around the plus sign Associative Law for Addition for Multiplication abc abc a b c a b c Move ( )’s around new plus sign 5 1 3 5 9 10 + 3 + 10 = 23 Move ( )’s around new times sign 5 1 3 9 4 20 27 = 540 Sect 1.2 Use the Commutative Law and Associative Law for Addition. 7 x 3 x 7 3 3 7 x x 7 3 3 7 x Use the Commutative Law and Associative Law for Multiplication. 4xy x4y y4 x x4 y y4x Use the Commutative Law for Addition and Multiplication. 7x 3 3 7 x 3 x7 Sect 1.2 Distributive Property ab c ab ac Separate by place values & add. 4x 3 5237 5200 30 7 1000 + 150 + 35 = 1185 42 x y 6 4x – 12 2a5b 3c 9d 1 8x – 4y + 24 10ab + 6ac – 18ad – 2a Factor using the Distributive Property 7x 7 y 7 x 7 y 12 x 8 y 4 4 3 x 4 2 y 4 1 7 x y 4 3x 2 y 1 GCF leftovers Sect 1.2 Terms vs Factors Term is any number, variable, or quantity being multiplied together. Be careful of the definition that terms are separated by plus or minus signs. Only if the ( )’s are simplified away! One term two terms three terms x 10 7 w 2ab 5 2ab 10a 2 2 x 3 y 4 Multiplying Factors are the number, variable or quantity being multiplied together. 2ab 5 The factors are 2, a, and (b – 5) Sect 1.3 Review: Natural Numbers = { 1, 2, 3, 4, 5, 6, …..} List factors of 18. The factors are 1, 2, 3, 6, 9, 18 Prime Numbers are Natural numbers that have 2 different factors, 1 and itself. {2, 3, 5, 7, 11, 13, 17, 19, 23, …} Composite Numbers are Natural numbers that have 3 or more factors. {4, 6, 8, 9, 10, 12, 14, 15, …} Notice that “1” is not in either set! Sect 1.3 List the prime factorization of 48. 48 2 2 2 2 3 Tree method 48 4 2 Always start with smallest prime numbers and work up to largest prime number. 12 6 2 2 2 Staircase Method 2 3 Division Rules 2: any even number 3: sum of the digits is divisible by 3 5: ends in 0 or 5 48 2 24 2 12 2 6 3 The prime number outside the upside down division boxes should be all the prime numbers. 48 2 2 2 2 3 Sect 1.3 Fraction notation. a numerator b denominator Fraction Properties Notation for 1 a 1 a Notation for 0 Undefined 0 0 b a undefined 0 Why do we use the undefined term? We have to define Multiplication and Division with the same numbers. Example 5 3 15 15 3 5 5 0 0 Multiplying by 0 and divide by 0 doesn’t 00 5 We start with 5 and finish with 5 when we multiply by 3 and divide by 3. return to the original value, not defined. Sect 1.3 Fraction multiplication. Tops together and bottoms together. 3 4 3 4 12 12 1 1 8 15 8 15 120 12 10 10 a c ac b d bd Another technique is to Simply first. Multiplicative inverse (Reciprocal) a b 1 b a 13 3 4 4 1 1 1 1 8 15 4 2 3 5 2 5 10 We don’t divide by fractions, but Multiply by the reciprocal of Fraction Division the fraction that we are dividing by. 1 1 a c a d ad 7 35 7 48 7 12 4 4 b d b c bc 12 48 12 35 12 7 5 5 a a a c ac 1 Multiplicative Identity b b b c bc Use this property to get common denominators. Sect 1.3 Simplify the fraction by multiplication rules. 15 5 3 3 40 5 8 8 36 12 3 3 24 12 2 2 9 9 1 1 72 9 8 8 Canceling errors! 4 1 1 42 2 23 3 2 1 Can’t cancel with addition or subtraction! Addition and Subtraction of Fractions (same Denominators) a c ac b b b 1 5 1 6 1 12 12 12 2 2 a c ac b b b 3 11 5 6 3 8 8 8 4 4 Sect 1.3 Addition and Subtraction of Fractions (with different Denominators) a c a d c b ad bc ad bc Rule that works every b d b d d b bd bd bd time, however, can a c a d c b ad bc ad bc create huge numbers! b d b d d b bd bd bd 7 11 7 12 11 8 84 88 172 4 43 43 8 12 8 12 12 8 96 96 4 24 24 96 We can work with smaller numbers and prior knowledge…staircase method. 7 11 4 8, 12 2, 3 8 12 Multiply the outsides for the LCD = 4(2)(3) = 24 Notice cross multiplying = 24 9 5 8 6 2 8, 6 4, 3 Multiply the outsides for the LCD = 2(4)(3) = 24 21 22 43 3 7 11 2 3 8 12 2 24 24 24 3 9 5 4 27 20 7 3 8 6 4 24 24 24 Sect 1.4 Positive and Negative Real Numbers Review: The Set of Numbers REAL NUMBERS Any number on the number line. IRRATIONAL NUMBERS Numbers that can’t be written as a fraction Examples : , 3 1.73205... 0 RATIONAL NUMBERS Numbers that CAN be written as a fraction Examples : 9, 34 ,0.3, 4 2 INTEGERS NUMBERS … -3, -2, -1, 0, 1, 2, 3, … WHOLE NUMBERS 0, 1, 2, 3, … NATURAL NUMBERS 1, 2, 3, … Less Than, Greater Than < > Less Than or Equal to, < 2.0310 ___ 2.0309 Greater Than or Equal to > To compare decimal numbers, both numbers need to have the same number of decimal places. Add a 0 to the end of the left number and compare place values until different. 10 > 9 To compare fractions, we need common denominators. Multiply the other denominators to the numerators and compare the products. 6 12 7 11 72 ___ 77 ___ 11 12 12 11 6 7 11 12 Sect 1.4 Positive and Negative Real Numbers Absolute Value The POSITIVE distance a number is away from zero on the number line. 5 5 7 7 7 5 units long -5 -4 -3 -2 -1 0 1 2 Convert a repeating decimal to fraction. 0.3 Step 1. Set the repeating decimal = x Step 2. Get the decimal point to the left of the repeating digits. Already done. Step 3. Get the decimal point to the right of the repeating digits. Multiply by 10’s to both sides of the equation. This moves the decimal point one place for each 10. Step 4. Subtract Step 3 – Step 2 and solve for x. x 0.3 1 3 x 0.3 10 x 10 0.33 10x 3.3 x 0.3 9x 3 x 3 9 1 3 Convert a repeating decimal to fraction. 0.63 Step 1. Set the repeating decimal = x Step 2. Get the decimal point to the left of the repeating digits. Already done. x 0.63 7 11 x 0.63 Step 3. Get the decimal point to the right 100 x 100 0.6363 of the repeating digits. Multiply by 10’s to 100 x 63.63 both sides of the equation. This moves the decimal point one place for each 10. x 0.63 Step 4. Subtract Step 3 – Step 2 and solve for x. 99x 63 x 63 99 7 11 Convert a repeating decimal to fraction. 0.16 Step 1. Set the repeating decimal = x Step 2. Get the decimal point to the left of the repeating digits. Multiply by 10. Step 3. Get the decimal point to the right of the repeating digits. Multiply by 10’s to both sides of the equation. This moves the decimal point one place for each 10. Step 4. Subtract Step 3 – Step 2 and solve for x. x 0.16 1 6 10x 1.66 10 10x 10 1.66 100x 16.66 10x 1.66 90 x 15 x 15 90 1 6 Sect 1.5 and 1.6 Add & Subtract sign numbers Add & Subtract with number line. 3 Step rule. Any two signed numbers. 1. Remove all double signs. a–(-b) a+b a+(-b) a–b 2. Keep the sign of the largest number ( absolute value ). +Large – small = Positive answer Small – Large = Negative answer 3. a. Same Signs Sum b. Different Signs Difference (subtract) +Large – small = + (Large – small) – Large + small = – ( Large – small) - a – b = - (a + b) + a + b = + (a + b) Sect 1.5 and 1.6 Add & Subtract sign numbers -12 + (-7) 1. Double signs -15 + 9 2 Sign of Largest number 1. Double signs NONE -12 – 7 = – 19 -16 – 18 1. Double signs NONE -32 – (-4) 23 + (-11) 1. Double signs 2 Sign of Largest number 2 Sign of Largest number 23 – 11 = + 12 = – 34 3 Same signs SUM 1. Double signs 2 Sign of Largest number 19 – (-7) 3 Different signs Difference LG - sm 3 Different signs Difference LG - sm 1. Double signs 2 Sign of Largest number 19 + 7 = + 26 3 Same signs SUM -9 + (-7) – (-4) + 3 – 8 – (-12) 1. Double signs = –6 3 Different signs Difference LG - sm 3 Same signs SUM -32 + 4 = – 28 2 Sign of Largest number Add all positive numbers 1st and negative numbers 2nd. -9 – 7 + 4 + 3 – 8 + 12 19 – 24 = – 5 2 Sign of Largest number Law of Opposites: a + (-a) = 0 Good to use this property when adding a long list of sign numbers…canceling is good! 3 Different signs Difference LG - sm Sect 1.5 and 1.6 Add & Subtract sign numbers Combine Like Terms Defn. 1. Must have the same variables in the individual terms. 2. The exponents on each variable must be the same. Identify the like terms. 7x + 3y – 5 + 2x – 9y – 8x + 10 x – terms y – terms Now Combine them. 7x + 2x – 8x constants + 3y – 9y – 5 + 10 x – 6y + 5 Combine Like Terms 2a + (- 3b) + (-5a) + 9b 2xy + 3x – 7y + 5 – 8x – 2 + y 2a – 3b – 5a + 9b 2xy + 3x – 7y + 5 – 8x – 2 + y – 3a + 6b 2xy – 5x – 6y + 3 Sect 1.7 Mult and Division of sign numbers 2 steps 1. Determine the sign. Even number of Negatives being multiplied or divided = Positive answer. Odd number of Negatives being multiplied or divided = Negative answers. 2. Multiply or divide the values. 3 2 5 14 3 360 1 2 10 3 4 1 3 2 7 1 20 4 514 3 36 Multiply by 0 rule. 310 15 110 372 Sign on the fraction rule. a a a b b b 0 Sect 1.8 Exponential Notation & Order of Operations Exponential notation is a short cut to writing out repetitive multiplication. Negative quantities are 6 aaaaaa a a 1 a defined as a -1 multiplied to the positive quantity. Simplify. 3 3 3 3 3 4 99 81 2x 34 3 3 3 3 3 2x 2x 2x 8x 3 99 81 2x 3 2 x x x 2x 3 34 1 34 1 3 3 3 3 1 9 9 81 Sect 1.8 Exponential Notation & Order of Operations Order of Operation P.E.MD.AS 1. P = ( )’s which means all grouping symbols. ( ), { }, [ ], | |, numerators, denominators, square roots, etc. 2. E = Exponents. All exponential expressions must be simplified. 3. MD = Multiply or Divide in order from Left to Right 4. AS = Add or Subtract in order from Left to Right 15 2 5 3 15 10 3 5 3 8 2 9 2 2 129 7 4 5 34 2 3 8 4 9 23 5 3 3 2 9 2 8 2 9 16 27 5 Top Bottom 129 7 4 5 34 2 3 122 20 81 8 24 20 89 44 44 89 Sect 1.8 Exponential Notation & Order of Operations Simplify When variables are present, remove ( )’s by the Distributive Property and Combine Like Terms. Include the sign 5x 9 24x 5 7 x 2 3 x 2 2 x 5x 7 x 2 3x 2 6 x 5 x 5x 9 8x 10 13x 1 10x 2 x 2, STO> button, X, enter Original Expression Our Answer x 7 y 5 1 x 7 y 5 x 7y 5 7 3x 2 7 3x 6 3x 4 x 2 3x 4x 2 x2 5t 2 2t 4t 2 9t 9t 2 11t 7 x 2 5x 5 8 7 x 2 5 x 3 7 x3 2 5 x3 1 8 3 3 3 1 3x 5t 2 2t 4t 2 9t 3 7 x3 2 5x3 3 2 x3 1