A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 + 5-7 + 10 + 15-4 E. : -8h + 10h - 1h - 15h -8h + 10h - 1h - 15h -0h - 5h II. Appling the Distributive Propert - Ever term inside the parentheses is multiplied b the term outside of the parentheses. E. 1: (9 " 4) # 9 " # 4 7 "1 E. : 4 (5 + 6) 4 " 5 + 4 " 6 0 5 + 4 III. Combining Like Terms AND the Distributive Propert (Problems with a Mi!) - Sometimes problems will require ou to distribute AND combine like terms!! E. 1: (4 " ) +1 # 4 " # +1 1 " 6 +1 5 " 6 E. : (1 " 5) " 9("7 +10) #1 " # 5" 9("7) " 9(10) 6 "15+ 6" 90 " 54 + 48 5
PRACTICE SET 1 Simplif. 1. 8! 9 + 16 + 1. 14 +! 15 +. 5n! (! 4n) 4.! (11b! ) 5. 10 q (16 + 11) 6.! ( 5! 6) 7. (18z! 4w) + (10z! 6w) 8. ( 8c + ) + 1(4c! 10)! 9. 9(6! )! (9 ) 10.! (! ) + 6(5 + 7) 6
I. Solving Two-Step Equations B. Solving Equations A couple of hints: 1. To solve an equation, UNDO the order of operations and work in the reverse order.. REMEMBER! Addition is undone b subtraction, and vice versa. Multiplication is undone b division, and vice versa. E. 1: 4 " = 0 + + 4 = 4 4 = 8 E. : 87 = "11 + 1 " 1 " 1 66 = "11 "11 "11 " 6 = II. Solving Multi-step Equations With Variables on Both Sides of the Equal Sign - When solving equations with variables on both sides of the equal sign, be sure to get all terms with variables on one side and all the terms without variables on the other side. E. : 8 + 4 = 4 + 8 " 4 " 4 8 = 4 + 4 " 4 " 4 4 = 4 4 4 = 6 III. Solving Equations that need to be simplified first - In some equations, ou will need to combine like terms and/or use the distributive propert to simplif each side of the equation, and then begin to solve it. E. 4 : 5(4 " 7) = 8 + 45+ 0 " 5 =10 + 45 "10 "10 10 " 5 = 45 + 5 + 5 10 = 80 10 10 = 8 7
PRACTICE SET Solve each equation. You must show all work. 1. 5! =. 140 = 4 + 6. 8 (! 4) = 196 4. 45! 70 + 15 = 60 5. 1 = 4(1! 9) 6. 198 = 154 + 7! 68 7.! 11 =! 5(! 8) + 6 8.! 7! 10 = 18 + 9. 1 + 8! 15 =! (! 8) 10.! ( 1! 6) = 1 + 6 IV. Solving Literal Equations - A literal equation is an equation that contains more than one variable. - You can solve a literal equation for one of the variables b getting that variable b itself (isolating the specified variable). E.1: =18, Solve for. = 18 = 6 E. : 5a "10b = 0, Solve for a. +10b =+10b 5a = 0 +10b 5a 5 = 0 5 + 10b 5 a = 4 + b 8
PRACTICE SET Solve each equation for the specified variable. 1. Y + V = W, for V. 9wr = 81, for w. d f = 9, for f 4. d + t = 10, for 5. P = (g 9)180, for g 6. 4 + 5h = 10 + u, for 9
C. Rules of Eponents Multiplication: Recall ( m )( n ) ( m+ n) = E: ( 4 )(4 5 )=(" 4)( 4 " 1 )( " 5 )=1 5 7 Division: Recall m ( m n)! n 5 5 4m j ' 4 $ ' m $ ' j $ = E: = 14m j % " =! 1 m j % m " % j "! &! #& #& # Powers: Recall ( m ) n ( m! n) = E: 4 1 4 9 1 (! a bc ) = (! ) ( a ) ( b ) ( c ) =! 8a b c 0 Power of Zero: Recall = 1,! 0 E: 0 4 4 4 5 = (5)(1)( ) = 5 PRACTICE SET 4 Simplif each epression. 15 5 m 1. ( c )( c)( c ). m. (k 4 ) 5 4. 0 4 7 5 d 5. ( q )( p q ) p 6. 45 z 5 z 10 7. (! t 7 ) 8. g 0 5 f 9. (4h k )(15k h ) 10. 4 6 1a b 6ab c 11. ( n m ) 4 1. ) 0 ( 1 1. (! 5a b)(ab c)(! b) 14. 4 0 ( ) 15. 4 ( )( ) 10
I. Reviewing the Distributive Propert D. Binomial Multiplication The distributive propert is used when ou want to multipl a single term b an epression. E 1: 8(5 8 " 5 40! 9) + 8 " (! 9)! 7 II. Multipling Binomials the FOIL method When multipling two binomials (an epression with two terms), we use the FOIL method. The FOIL method uses the distributive propert twice! FOIL is the order in which ou will multipl our terms. First Outer Inner Last E. 1: ( + 6)( + 10) FIRST OUTER First " ------> ( + 6)( + 10) Outer Inner 10 -----> 10 6 ------> 6 INNER LAST Last 6 10 -----> 60 + 10 + 6 + 60 + 16 + 60 (After combining like terms) 11
Recall: 4 = 4 4 = E. ( + 5) ( + 5) = ( + 5)(+5) Now ou can use the FOIL method to get a simplified epression. PRACTICE SET 5 Multipl. Write our answer in simplest form. 1. ( + 10)( 9). ( + 7)( 1). ( 10)( ) 4. ( 8)( + 81) 5. ( 1)(4 + ) 6. (- + 10)(-9 + 5) 7. (- 4)( + 4) 8. ( + 10) 9. (- + 5) 10. ( ) 1
E. Factoring I. Using the Greatest Common Factor (GCF) to Factor. Alwas determine whether there is a greatest common factor (GCF) first. E. 1 4! + 90 In this eample the GCF is. So when we factor, we have (! 11 + 0). Now we need to look at the polnomial remaining in the parentheses. Can this trinomial be factored into two binomials? In order to determine this make a list of all of the factors of 0. 0 0 1 0 15 10 5 6-1 -0 - -15 - -10-5 -6 Since -5 + -6 = -11 and (-5)(-6) = 0 we should choose -5 and -6 in order to factor the epression. The epression factors into (! 5)(! 6) Note: Not all epressions will have a GCF. If a trinomial epression does not have a GCF, proceed b tring to factor the trinomial into two binomials. II. Appling the difference of squares: a! b = ( a! b)( a + b) E. 4 "100 ( ) 4 " 5 ( )( + 5) 4 " 5 Since and 5 are perfect squares separated b a subtraction sign, ou can appl the difference of two squares formula. 1
PRACTICE SET 6 Factor each epression. 1. + 6. 4 a b! 16ab + 8ab c.! 5 4. n + 8n + 15 5. g! 9g + 0 6. d + d! 8 7. z! 7z! 0 8. m + 18m + 81 9. 4! 6 10. 5k + 0k! 15 14
F. Radicals To simplif a radical, we need to find the greatest perfect square factor of the number under the radical sign (the radicand) and then take the square root of that number. E. 1: 7 6 " 6 E. : 4 90 4 " 9 " 10 4 " " 10 1 10 E. : 48 16 4 OR E. : 48 4 1 1 4 " " This is not simplified completel because 1 is divisible b 4 (another perfect square) 4 PRACTICE SET 7 Simplif each radical. 1. 11. 90. 175 4. 88 5. 486 6. 16 7. 6 500 8. 147 9. 8 475 10. 15 9 15
G. Graphing Lines I. Finding the Slope of the Line that Contains each Pair of Points. Given two points with coordinates ( 1, 1) and (, ) the line containing the points is! m = 1.! E. (, 5) and (4, 1) E. (-, ) and (, ) 1! 5! 4! 1 m = = =! m = = 4!! (! ) 5 1 The slope is -. The slope is 5 1, the formula for the slope, m, of PRACTICE SET 8 1. (-1, 4) and (1, -). (, 5) and (-, 1). (1, -) and (-1, -) 4. (, -4) and (6, -4) 5. (, 1) and (-, -) 6. (5, -) and (5, 7) 16
II. Using the Slope Intercept Form of the Equation of a Line. The slope-intercept form for the equation of a line with slope m and -intercept b is E. =! 1 E. =! + 4 Slope: -intercept: -1 Slope:! -intercept: 4 = m + b. Place a point on the -ais at -1. Place a point on the -ais at. Slope is or /1, so travel up on Slope is -/4 so travel down on the the -ais and over 1 to the right. -ais and over 4 to the right. Or travel up on the -ais and over 4 to the left. PRACTICE SET 9 1 1. = + 5. =! Slope: -intercept: Slope: -intercept: 17
. =! + 4 5 4. =! Slope: Slope: -intercept: -intercept 5. =! + 6. = Slope: Slope: -intercept: -intercept 18
Algebra Summer Review Packet Student Answer Ke A. Simplifing Polnomial Epressions PRACTICE SET 1 1. 4 +.! 15 + 7 +. 9 n! 4.! b + 6 5. 160 q + 110q 6.! 5 + 6 7. 74 z! 4w 8. 56 c! 117 9.! 7 + 54! 9 10.! + 1 + 4 B. Solving Equations PRACTICE SET 1. = 7. = 6. = 9. 5 4. = 1 5. =. 5 6. = 16 7. = 19 8. =!. 8 9. = 9. 5 10. = 0 PRACTICE SET 1. V = W! Y. w = 9 r. f = 9 " d " = " + d 4. 10! t 10 t = =! d d d 5. g = P + 160 180 = P 180 + 9 6. 9 + u + 5h = 4 6
C. Rules of Eponents PRACTICE SET 4 8 1. c. 4. 1 5. 1 m. 0 k p 11 q 7 6. 9 9z 7. 1! t 8. f 9. 8 60h k 5 10. a b c 4 11. 8 4 81m n 1. 1 4 1. 0 a b c 14. 4 15. 4 7 4 D. Binomial Multiplication PRACTICE SET 5 1. +! 90.! 5! 84.! 1 + 0 4. + 7! 648 5. 8 +! 6. 50! 100 + 18 7. "6 " 0 " 16 8. + 0 + 100 9. 5! 10 + 10. 4! 1 + 9 E. Factoring PRACTICE SET 6 1. ( + ). 4ab ( a! 4b + c). (! 5)( + 5) 4. ( n + 5)( n + ) 5. ( g! 4)( g! 5) 6. ( d + 7)( d! 4) 7. ( z! 10)( z + ) 8. ( m + 9) 9. 4 (! )( + ) 10. 5 ( k + 9)( k! ) 7
F. Radicals PRACTICE SET 7 1. 11. 10. 5 7 4. 1 5. 9 6 6. 8 7. 60 5 8. 1 9. 40 19 10. 5 5 G. Graphing Lines PRACTICE SET 8 1... " 1 4. 0 5. 1 6. undefined PRACTICE SET 9 1. Slope: -intercept: 5. Slope: 1 -intercept: - 8
. =! + 4 4. =! 5 Slope:! Slope: - 5 -intercept: 4 -intercept 0 5. =! + 6. = Slope: -1 Slope: 1 -intercept: -intercept 0 9
PRACTICE SET 10 1... 4. 5. 6. 0