Sect 1.1 Algebraic Expressions Variable Constant Variable Expression Evaluating the Expression Area formula Perimeter Consist of variables and/or numbers, often with operation signs and grouping symbols. Any symbol that represents a number…letters or A value that never changes. An expression that contains a variable. To evaluate an expression, we substitute a value in for each variable in the expression and calculate the result. 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 subtracted from difference of minus less than decreased 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 2 was subtracted from the number difference of two numbers 8 minus some number 9 less than the number The number decreased by 10 n + 5 x m r + 20 y + 3 w - 2 a - b 8 - c d - 9 f - 10

Sect 1.1 multiplied by product of times twice of divided by quotient of divided into ratio of per the number multiplied by 4 the product of two numbers 13 times some number twice the number half of the number 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

Sect 1.1 Four less than Joe’s height in inches. Eighteen increased by a number. A day’s pay divided by eight hours. Half of the pallet. Seven more than twice a number. Six less than the product of two numbers. Nine times the difference of a number and 3. Eighty five percent of the enrollment. Twice the sum of a number and 3. The sum of twice a number and 3. h – 4 ab – n9(m – 3) p/8 p/8 85%(e) = 0.85(e) 1/2 p 1/2 p 2(x + 3) 2x + 7 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. 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. True 32 = 32False 5 = 6We don’t know the value of x = = 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? Twice the difference of a number and 4 is 24. Three times a number plus seven is the same as the number less than one. 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? x+478= ( ________ – _________ )x4= 24 3x3x+ 7=x – 1 “than” makes the terms switch around the minus sign 1666 =P – = P

Sect 1.2 Equivalent expressions ,, and 3(4) Laws that keeps expressions equivalent. Commutative Law for Addition for Multiplication Associative Law for Addition for Multiplication switch around the plus sign switch around the times sign Move ( )’s around new plus signMove ( )’s around new times sign = = 540

Sect 1.2 Use the Commutative Law and Associative Law for Addition. Use the Commutative Law and Associative Law for Multiplication. Use the Commutative Law for Addition and Multiplication.

Sect 1.2 Distributive Property Factor using the Distributive Property Separate by place values & add = x4x– 12 8x8x – 4y ab + 6ac– 18ad– 2a 7 GCFleftovers 4

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 Factors are the number, variable or quantity being multiplied together. Multiplying The factors are 2, a, and (b – 5)

Sect 1.3 Review: Natural Numbers = { 1, 2, 3, 4, 5, 6, …..} List factors of 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! The factors are 1, 2, 3, 6, 9, 18

Sect 1.3 List the prime factorization of 48. Tree methodStaircase Method Division Rules 2: any even number 3: sum of the digits is divisible by 3 5: ends in 0 or 5 Always start with smallest prime numbers and work up to largest prime number. The prime number outside the upside down division boxes should be all the prime numbers.

Sect 1.3 Fraction notation. Fraction Properties Notation for 1Notation for 0Undefined numerator denominator Why do we use the undefined term? We have to define Multiplication and Division with the same numbers. Example We start with 5 and finish with 5 when we multiply by 3 and divide by 3. Multiplying by 0 and divide by 0 doesn’t return to the original value, not defined.

Sect 1.3 Fraction multiplication. Multiplicative inverse (Reciprocal) Fraction Division Multiplicative Identity Tops together and bottoms together. Another technique is to Simply first. We don’t divide by fractions, but Multiply by the reciprocal of the fraction that we are dividing by. Use this property to get common denominators.

Sect 1.3 Simplify the fraction by multiplication rules. Canceling errors! Addition and Subtraction of Fractions (same Denominators) Can’t cancel with addition or subtraction!

Sect 1.3 Addition and Subtraction of Fractions (with different Denominators) Rule that works every time, however, can create huge numbers! We can work with smaller numbers and prior knowledge…staircase method. Multiply the outsides for the LCD = 4(2)(3) = 24 Multiply the outsides for the LCD = 2(4)(3) = 24 Notice cross multiplying = 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 RATIONAL NUMBERS Numbers that CAN be written as a fraction 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 Greater Than or Equal to, 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.

Sect 1.4 Positive and Negative Real Numbers Absolute Value The POSITIVE distance a number is away from zero on the number line units long Convert a repeating decimal to fraction. 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.

Convert a repeating decimal to fraction. 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.

Convert a repeating decimal to fraction. 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.

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. 2. Keep the sign of the largest number ( absolute value ). 3. a. Same Signs Sum b. Different Signs Difference (subtract) a + ( - b ) a – b a – ( - b ) a + b +Large – small = Positive answerSmall – Large = Negative answer - a – b = - (a + b) + a + b = + (a + b) +Large – small = + (Large – small) – Large + small = – ( Large – small)

Sect 1.5 and 1.6 Add & Subtract sign numbers (-7) – (-11) -32 – (-4)19 – (-7) -9 + (-7) – (-4) + 3 – 8 – (-12) Law of Opposites: a + (-a) = 0 3 Same signs SUM 2 Sign of Largest number1. Double signs -12 – 7= –19 Add all positive numbers 1 st and negative numbers 2 nd. 1. Double signs NONE 3 Different signs Difference LG - sm 2 Sign of Largest number 1. Double signs -9 – – = –6 1. Double signs NONE 3 Same signs SUM 2 Sign of Largest number = –34 2 Sign of Largest number1. Double signs 23 – 11= Different signs Difference LG - sm 1. Double signs = –28 2 Sign of Largest number 3 Different signs Difference LG - sm 3 Same signs SUM 2 Sign of Largest number1. Double signs = – 24 2 Sign of Largest number = –5 3 Different signs Difference LG - sm Good to use this property when adding a long list of sign numbers…canceling is good!

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 Now Combine them. Combine Like Terms 2a + (- 3b) + (-5a) + 9b2xy + 3x – 7y + 5 – 8x – 2 + y x – terms 7x + 2x – 8x y – terms + 3y – 9y constants – x – 6y + 5 2a – 3b – 5a + 9b – 3a + 6b 2xy + 3x – 7y + 5 – 8x – 2 + y 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. Multiply by 0 rule. Sign on the fraction rule. 2341

Sect 1.8 Exponential Notation & Order of Operations Exponential notation is a short cut to writing out repetitive multiplication. Simplify. Negative quantities are defined as a -1 multiplied to the positive quantity.

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

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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 Original Expression Our Answer 2, STO> button, X, enter