GEORGE RANCH HIGH SCHOOL ALGEBRA I PAP SUMMER PREP PACKET 2016
Integer Addition, Subtraction, Multiplication, Division BASIC DEFINITIONS: INTEGERS Positive and Negative numbers (and zero) whose decimal digits are zeros. ABSOLUTE VALUE Distance from zero on a number line. OPPOSITES Two numbers the same distance from zero on a number line but on different sides of zero. INTEGER ADDITION: - Do the integers have the same sign? YES -ADD their absolute values -Keep the common sign absolute value. NO -SUBTRACT their absolute values -Keep the sign of the integer with the larger INTEGER SUBTRACTION: - Add the opposite. How? Step 1: Keep the first integer the same Step 2: Change the subtraction symbol to addition Step : Change the sign (to its opposite) of the sign that follows the subtraction symbol Step 4: Follow the rules of addition above. INTEGER MULTIPLICATION and DIVISION: - Do the integers have the same sign? YES -MULTIPLY or DIVIDE their absolute values -Your answer will always be positive NO -MULTIPLY or DIVIDE their absolute values -Your answer will always be negative 2
Name Integer Operations: Add/ Subtract Alg. 1 Intro Packet Simplify by performing the operation. Do not use a calculator: 1) 4 + ( 7) 2) + ( 1) ) 6 + 9 1) 2) ) 4) 4) ( 2) + ( ) 5) + 11 6) ( 6) + ( 10) 7) 7 + ( 7) 8) 6 + ( 4) 9) 1 + ( ) 5) 6) 7) 8) 9) 10) 4 + 7 11) + ( 9) 12) 2 + 9 10) 11) 12) 1) 1) 12 + ( 2) 14) 5 + 10 15) 6 + ( 11) 14) 15) Simplify by performing the operation. Do not use a calculator: 1) 4 7 2) ( 1) ) 6 9 1) 2) ) 4) 4) 2 5) 1 ( ) 6) 6 10 7) 7 7 8) 6 ( 4) 9) 11 5) 6) 7) 8) 9) 10) 4 7 11) ( 9) 12) 2 9 1) 12 ( 2) 14) 5 10 15) 6 11 10) 11) 12) 1) 14) 15)
Name Integer Operations: Mult. & Divide Alg. 1 Intro Packet Simplify by performing the operation. Do not use a calculator: 1) 5 2 2) ( 7)( ) ) ( 10)(5) 1) 2) ) 4) 4) 12 5) 9( 2) 6) ( 1)( ) 5) 6) 7) 8) 9) 7) 6 (8) 8) ( 10)( 5) 9) 5 8 10) 2 7 11) 2( 2) 12) ( 11) 7 10) 11) 12) 1) 1) 5 ( 5) 14) 42 ( ) 15) 28 7 14) 15) 16) 17) 18) 16) 14 2 17) 9 ( ) 18) 1 19) 48 8 20) 10 5 21) 40 20 19) 20) 21) 22) 2) 24) 22) 21 ( 7) 2) 2 ( 2) 24) 100 2 4
Name Alg. 1 Intro Packet MIXED INTEGER REVIEW Add, subtract, multiply or divide the following integers. Do not use a calculator: 1) 2 + ( 7) 2) 4 ( ) 1) 2) ) 4) ) 15 4) ( 9)( 7) 5) 6) 7) 8) 5) ( 6) 6) 4 5 7) 7 + 8) 2 + ( 6) 9) 10) 11) 12) 1) 14) 15) 16) 17) 18) 9) 4 5 10) 12 ( 7) 19) 20) 11) 15 + 7 12) 11 5 1) 8 ( 21) 14) ( )( 9) 15) 24 4 16) 9 + ( 1) 17) ( 2)( 25) 18) 50 0 19) 56 8 20) 2 + ( 7) 5
ALGEBRAIC PROPERTIES These basic algebraic properties, which do not affect an expression s outcome, are categorized by: -Movement of terms -Grouping of terms -Special results from performing certain operations. MOVEMENT OF TERMS: COMMUTATIVE PROPERTY (To commute means to move ) of Addition: of Multiplication: a + b b + a a b b a 5 + 4 4 + 5 5 4 4 5 + 7 7 + ( ) 7 7 ( ) x + 6 6 + x x 6 6 x GROUPING OF TERMS: ASSOCIATIVE PROPERTY (To associate implies who is grouped together) of Addition: of Multiplication: a + ( b + c) ( a + b) + c a ( b c) ( a b) c 5 + (4 + ) (5 + 4) + 5 (4 ) (5 4) + [7 + ( 1)] ( + 7) + ( 1) [7( 1)] ( 7) ( 1) x + ( 6 + y) ( x + 6) + y x ( 6y) ( x 6) y SPECIAL RESULTS FROM PERFORMING CERTAIN OPERATIONS: DISTRIBUTIVE IDENTITY PROPERTY PROPERTY a ( b ± c) ab ± ac of Addition: of Multiplication: a + 0 a a 1 a 5(4 + ) 5 4 + 5 [7 + ( 1)] (7) + ( )( 1) 5 + 0 5 5 1 5 x( 6 + y) x 6 + x y + 0 1 x + 0 x x 1 x *Distributive Property in reverse, ab ± ac a( b ± c), can still be called the Distributive Property, but is more commonly known in algebra as FACTORING (through GCF). 6
Name Algebraic Properties Alg. 1 Intro Packet Match the expression change with the algebraic property that justifies it: 1) 0 + = 2) + 6 = 6 + [A] Commutative Property of Addition ) ( 2 5)8 = 2(5 8) 4) 1 52 = 52 [B] Commutative Property of Multiplication 5) ( 1+ 2) + = 1+ (2 + ) 6) 7 8 = 8 7 [C] Associative Property of Addition 7) 5 + 1 = 1+ 5 8) 4( 7) = (4 ) 7 [D] Associative Property of Multiplication 9) 2 7 = 7 2 10) ( xy ) + z = z + ( xy) [E] Distributive Property 11) 4( + 7) = 4 + 4 7 12) 2 7 = 7 2 [F] Additive Identity ( Identity Property of Addition ) 1) 2) ) 4) 5) 6) 7) 8) 9) 10) 11) 12) 1) *1) 18 x + 9 = 9(2x + 1) [G] Multiplicative Identity ( Identity Property of Multiplication ) Replace the question mark with the missing information: 14) Associative Property of Addition: 6 + (7 + d ) (? + 7) + d 15) Associative Property of Multiplication: 5 [ ( 4) ] [ 5? ] ( 4) 16) Distributive Property: 7( x + 5) =? x +? 5 17) Additive Identity (Identity Property of Addition):? + 0 = m 18) Commutative Property of Addition: jk + mn = mn +? 14) 15) 16) 17) 18) 7
Name Distributive Property Alg. 1 Intro Packet Use the Distributive Property to simplify the following expressions: 1) 5 ( x + 4) 2) 5 ( x + 4) 1) 2) ) 8( x 2) 4) 8(x 4) 5) 2(5 x) 6) 2(5 + 2b) 7) 4(2x 5) 8) 4(2b 5) ) 4) 5) 6) 7) 8) Simplify Like Terms: 9) 4 + 2x + 20 x 10) 11 x 7x + 44 + 6 9) 10) 11) m 12 + 2 12) 5h + 4h + 75 h 11) m 12) 1) 14) 1) 8 8 12x + 8 x 14) 21 18x + 7 15) 16) 15) 5) 6 4 ( b + 16) 2 y 6( y + 4) 8
EXPRESSIONS and EQUATIONS EXPRESSION Collection of numbers, operations, and variables. Examples: 2 + 7 x a + 4 5 c 7 x 8 5 EQUATION Two expressions separated by an equals sign. Examples: 2 + 7 = 12. 4 x a + 4 = 7 5 c = 40 v 12. 4 EVALUATING EXPRESSIONS: To evaluate an expression, substitute/replace the variable with the number and simplify: Examples: + 4 w c 0 5 (6 a if a = 7 5 c if = 6 ( 7) + 4 ) 11 0 10 if w = 0 SOLVING EQUATIONS: OPEN SENTENCE Equation with at least one variable to be solved. Equations can be True, False, or Open. 18 + 2 = 50 True 45 15 = 25 False 12 = 4 + x Open REPLACEMENT SET Collection of numbers that are substituted in for the variable(s). Example: Solve for 2 x = x + using replacement set { 0,1, 2, } Solutions: 2(0) (0) +?? = 2(1) = (1) + 2(2) = (2) + 2() = () + 0 2 4 4 5 6 = 6 NO NO NO YES Solution Set: {}?? 9
To solve an equation without a replacement set provided, performing the inverse operation of the one(s) in the original equation helps to isolate the variable. Examples: w w + 5 = 12 5 = 5 = 7 y y 4 = 18 + 4 = + 4 = 22 4x = 20 4 20 = 4 4 x = 5 f x f [ ] = [ 8] f = 8 = 24 WRITING EQUATIONS: Addition Words Subtraction Words Multiplication Words Division Words Sum Difference Product of Quotient Add Subtracted from Multiplied by Divided by More than Less than Times Increased by Fewer than Double Decreased by Triple Is means equal to. Of often means multiplication. To write an equation, use your key words to translate the phrases into algebraic expressions/equations. Examples: Five more than g is 4 72 is one-sixth of y 1 g ) 72 = 6 y y (or also 72 = ) 6 Solutions: 5 + g = 4 (or also + 5 = 4 1 5 = 5 6 ( 72) = 6( y) 6 g = 29 42 = y 10
Name One-Step Equations: Addition/Subtraction Alg. 1 Intro Packet Solve each equation by isolating the variable. Show all work: y 2) t + 8 = 1 ) + r = 11 1) + 7 = 5 4) 6 + p = 9 5) 2 = x + 6 6) 1 = w + 7 7 8) 4 = 1+ z 9) = 9 + j 7) = 11+ n Solve each equation by isolating the variable. Show all work: y 11) t 5 = 1 12) v 4 = 11 10) 4 = 5 1) 1 = x 7 14) 10 = x 8 15) 10 = x 8 16) p 6 = 9 17) 2 = x 1 18) 5 = a ( 11) 11
Name One-Step Equations: Mult./ Division Alg. 1 Intro Packet Solve each equation by isolating the variable. Show all work: y 2) 4t = 16 ) 6 x = 12 1) = 15 4) 7 x = 77 5) 77 = 7x 6) 72 = 0.5x 7) 5 p = 90 8) 2b = 10 9) 10 = 5w Solve each equation by isolating the variable. Show all work: 1 1 10) 4 = 2 2 = 1 y 11) t 5 12) x = 11 1 1) 5 = 5 x x 14) 5 5 = x 15) = 5 5 1 16) 8 = 6 p 17) x ( 2) = 1 18) 1 = w 4 1 12
Name One-Step Equations: Mixed Review 1 Alg. 1 Intro Packet Solve each equation by isolating the variable. Show all work: b g 2) 24 + y = 61 ) = 5 1) = 15 6 4) 7 + = 17 y 5) c ( 0) = 12 6) 4b = 20 n 7) 2 = 9 t 8 = 9) 48 = 16g 8) 10 n 11) 4 = 6 + y 12) 6 = b 12 10) + 8 = 5 n 1) = 1 and 8 is negative. 14) x = 15) The sum of a number 1
Name One-Step Equations: Mixed Review 2 Alg. 1 Intro Packet Solve each equation by isolating the variable. Show all work: 16) = 27 b c 17) 12 8 = 18) 7 r = 49 x 20) 4b = 60 21) b + 4 = 4 19) 1 = 19 22) 16 = 78 + y 2) x 25 = 44 24) b 15 = 9 25) 0 h = 150 26) c ( 5) = 28 27) x ( 4) = 14 x 29) + x = 0) The difference of 1 28) 6 = a number and 1 is. 14
Name Writing & Solving Equations Alg. 1 Intro Packet A) Write and equation and B) Solve for the missing variable. Show all work: 1) Nine more than a number is four. 2) Ten less than a number is eight. ) The product of negative five and a number. 4) One-fifth of a number is negative three. is two-hundred fifteen. 5) A number decreased by sixteen is negative 6) Four less than a number is eight. twenty six. 7) Four copies of a book cost $44. Find the price 8) Jen added $150 to her savings account. Her of one book. balance is now $525. How much was it before? 9) Terri is 60 inches tall. This is 24 inches more 10) The perimeter of a square is 60 inches. Find than Kevin s height. How tall is Kevin? the length of each side. 15
EXPONENTS Exponents show repeated multiplication in shorthand form. An exponent tells us how many times to multiply the base by itself. EXAMPLES: 5 5 5 5 In this problem, 5 is called the base and is its exponent. 4a 4 a a a Note in this problem that is the exponent of base a, not the 4. (4a) 4a 4a 4a However in this example the 4 is included in the repeated multiplication. Both 4 and a are bases to exponent. -Writing an expression as just a base with its exponent is called writing it in exponential notation, such as 4 a a a becoming 4a in exponential notation. ORDER OF OPERATIONS An order has been agreed upon to which operations are performed before others. Several shortcut ways to remember this ranking system have been developed, with most popular being PEMDAS, or Please Excuse My Dear Aunt Sally. However, note that although there are six operations, two ranks of order have two operations in them. Step #1: Parentheses Compute within any grouping symbol first, if available. Step #2: Exponents Compute powers next, if available. Step #: Multiply or Divide Compute in order from left to right. Step #4: Add or Subtract Compute in order from left to right. - If more than one grouping symbol exists within a problem, such as (parentheses), [brackets], or {braces}, work from the inside out. EXAMPLES: Evaluate the following by using the Order of Operations: 4 2 x if x = 2 m + ( 5) if m = 6 n + 2 if n = 2 4 (2) 2 2 2 2 16 2 (6) 6 6 + ( 5) 6 + ( 5) 1 ( 2) ( 2)( 2)( 2) + 2 8 + 2 6 (a) if a = 2 a if a = 2 [(2)] [6] 6 6 6 216 2 24 8 2 16 4 + 16 4 + 9 4 + 9 1 1+ [4 (17 + 2) 2 ] 1+ [4 40 2 ] 1 + [4 40 8] 1 + [4 5] 1+ [ 1] 0 16
Name Order of Operations, page 1 Alg. 1 Intro Packet Find the answer by applying the Order of Operations. SHOW EACH STEP! Use example 1 as a guide: 2) 6 2 ( 5) ) 6 8 8 60 15 14 1 29 1 0 1) 15 7 2 1 4) 2 8 5 + 6 5) 20 5 4( 8) 6) 2 4 + 5 + 24 7) 10 12 12 ( 2) + 6 + 8) 6 + 2 8 + 9) 18 4 2 + 8 10) 7 + 8 2 8 + 6 11) 2 5) 12) ( 14 + 4) 5 6 ( 4 14) ( 2 + 1) 6 18 15) (8 6) 2 6 1) (2 4) 5 + 17
Name Order of Operations, page 2 Alg. 1 Intro Packet Find the answer by applying the Order of Operations. SHOW EACH STEP! 17) 49 7 8 + 4( + 2) 18) 2 (5 8) + 6 + 8 16) ( 8) 11+ 2 5 (6 + 8 + 18) 4 + 20) ( 4)( 9) + 6 ( + 4) 19) 0 8 Substitute and simplify each expression 21) 5 x when = 2 x 22) a 2 + 5 when a = 5 2) 2 + n when n = 5 24) b 4 5 when b = 25) w 11 when w = 5 26) x y & y = 6 when x = 2 18
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