Bridge to Algebra II.

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1 Regional Institute for Children & Adolescents Bridge to Algebra II Summer Pre-View Packet DUE THE FIRST DAY OF SCHOOL The problems in this packet are designed to help you review topics from previous Mathematics courses that are important to your success in Bridge to Algebra II. Show all work that leads you to each solution on separate sheets of paper. You may use your notes from previous mathematics courses to help you. You must do all work without any help from another person. Additional copies of this packet may be obtained from the Main Office or printed from the school's website: All work should be completed and ready to turn in on the FIRST DAY of school. ENJOY YOUR SUMMER!! WE ARE LOOKING FORWARD TO SEEING YOU IN THE FALL!! Student Name: Date:

2 NEe Algebra 2 Summer Review Name SHOW' ALL WORK ON A SEPARATE SHEET OF PAPER. I. Solve for x: 1) -4(3 - x) = 2(x + 6 ) 2)3x-2(x+1)=0 II. Solve the following systems of equations: 1) 5x + 4y = 6-2x -3y = -1 2)-2x+y=8 y = -3x - 2 Ill. Factoreach of the following polynomials: 1) x 2 - X - 72? 2) a- + 20a + 64 ") ~ ~.) m n - m n + zorn? 6) 2x-y - 4xy - 30y IV. Solve the following quadratic equations: 1)(2x + 1)(x + 3) = 0 2) p2 + 6p = 0 3) 1'2 + lor + 9 = 0 4)x:2=16 v. Determine each of the following: 1) Find a formula for the area of a rectangle with I = 2x + 3 and w = x - :2 2) Find a formula for the area of a square with s = 2x + 5 3) The area of a square with side 2x - 1 is 49. Find x. 4) Find the diagonal of a rectangle with! = 40 and w = 55. 5) The length of each leg of an isosceles right triangle is 4 cm. What is the length of the hypotenuse? VI. Simplify eachof the following: 1) (_3x 2 + 4x - 7) + (2x 2-7x + 8) 4 '" '1.1 i "') 2) (39a - 4a.J + 2a- - a - 7) - (loa + 3a' - 2a- -a + 8) 1

3 NEC Algebra 2 Summer Review 3) (3x + 7)(2x + 5) 5) (3x 2 + X - 1)(2x - 3) 6) (8a 3 b 2 )(2a 4 b 5 ) 10) (x + 6)2 VII. Graph each of the following on graph paper or create your own grid ) Y = - - x + 4 2) y = (x - 2) ) y = IxI 4 VIII. Given the following matrices, determine A =[ ~ ~3] C = [0 5] 1) A + B 2)A-B 3) -2C IX. Solve the following quadratic equations. using the quadratic formula: 1)2x~+3x-1=O -b±-!b 2-4ac x= a ~ 2) 3x- - 8x = -2, 3) 3x~ = 7-2x X. Answer each of the following concerning linear equations. 1) Determine the slope of the line containing the points (6, -2) and (-1,5). 2) Determine an equation for a line with slope Y2 and y-intercept at (0, -3). 3) Determine an equation for a line parallel to y = -3x +4, containing the point (2, 1).,.. 2

4 Add and Subtract Mixed Numbers When adding mixed numbers, we ca.n add the whole numbers and the fractions separately, then. "rsimpury h ~ - the answer. For example:. I j 4-+2-= 4-+2-= 6-=6+1-= 7-= When subtracting mixed numbers, we subtract the whole numbers and the fractions separately, then simplify the answer. For example: ~3 ]5 18 _15 _3 _1 /--2- =7--J-=)- =) = = = j Note: regrouping needed in order to subtract Exercises: Solve in lowest terms. No Calculators! SHOW.ALL WOR..K. Use a separate sheet of paper (if needed) and staple to this page ;::+8-::= ) / ") '7 ~ I = 10 9 I / 9--2-= 15 i2 6. _ 8 ~3 11--/-= ' Algebra 1 Page 4 Summer 2006

5 Multiply 8IDla Divide Mixed Numbers To multiply mixed numbers, we can first convert the mixed numbers into improper fractions. This is done my multiplying the denominator by the whole number part of the mixed number and them adding the numerator to this product. This sum is the numerator of the improper fraction. The denominator of the improper fraction is the same as the denominator of the mixed number For example: 3- leads to = 17, so 3- = -. ~ Once the mixed numbers are converted into improper fractions, we multiply and simplify just as j with regular fractions. For example: 5-& 3- = - Q- = -- = J 8- = ] To divide mixed numbers, we must convert t improper fractions then multiply by the reciprocal ) 3 of the second fraction and simplify. For example: 2- -':-3- = - -':- = = - = to 20 4 Exercises: Solve in lowest terms. ~10 Calculztors! SHOW ALL WORK. Use a separate sheet of paper (if needed) and staple to ~his page e6-= 3 5., ;). 1 7-G6 = 8 j 5 4--,:--= :--,:-~-::= ;) r =J, ;..1-= ~ l l,., / ':-1- = Algebra 1 Page 5 Summer 2006

6 Squares and Square Roots Exponents are a v,lay to represent repeated multiplication, so that 3 4 means 3 multiplied four times, or , which equals &1. In this example, 3 is the base and 4 is the power. Roots are the base numbers that correspond to a given power, so the square (referring to the power of 2) root of 81 is 9 because 9. 9 = 81 and the fourth root of 81 is 3 because is 81. if;;,where n is the root index and x is the radicand Exercises: Evaluate. ~ -LL.3 2. (-4i = = -'. 4.,//0-5.,\,289= 6. V6d =, VI"; J25-J6 = 11!.J..,,., 1.J }13.C;6-2)- Algebra 1 Page 6 2t.:IEDer :2006

7 Laws of Exponents There are certain rules when dealing with exponents that we can use to simplify problems. are'. Ad- _ mng " po\ver '" s a m - a - a Multiplying powers (am t =.: Subtracting po-wers am -=a an n - m+n «: m-n They Negative powers To the zero power Here are some examples of problems simplified using the above powers: 7 7 ~,' S _.;::5.a.. b (~5) ,:5-i- ~, =..,. ~,3 = ~,' 2..,. /-4 = 44 - = Exercises: Simplify the following problems using exponents (Do not multiply out) I. 8. o ~O Algebra 1 Page 7 Summer 2005

8 Summer r.iathernatics Packet Find Percent of a Number To determine the percent of a number, we must first convert the percent into a decimal by dividing by 100 (which can be short-cut by moving the decimal point in the percentage two places to the left), then multiplying the decimal by the number. For example: 4.5% of240 = 4.5%. 240 = = 10.8 Exercises: Solve for n. SHO\V ALL WORK. Use a separate sheet of paper (if needed) and staple to this page % of 450 = n % of 42 = n % of321 = n 4. 15% of 54 = n % of 320 = n % of 64 = n 7. 95%of568=n %of38=n %of348=n ~1(,of 488 = n % of 750 ==n % of 42 = n % of 78 = n 14. JAto of!r80 = n 15. O.lO c ioof435==n % of 54 =:l Algebra 1 Page 8 Summer 2006

9 Solve Problems Using Percents When solving percent problems, we apple the rules for finding percent of a number in realistic situations. For example, to ding the amount of sales tax on a $ item if the tax rate is 5%, we find 5% of 450 ( = 22.5), and then label our answer in dollars, getting $ Exercises: Solve the following items. SHOW ALL WORK. Use a separate sheet of paper (if needed) and staple to this page. 1. Susie has just bought a pair of jeans for $49.95, a sweater for $24.50, and a jacket for S85.9~. The sales tax is 5%. What is her total bill? Jack bought a set of golf ciubs for and received a rebate of :4%. the rebate? How much was A construction menacer calculates it will cost $ for mzterials :01 her next project.... '" - - She mus: add in.12.5% for SC_2.p am! extras. What will 'De the total cost? 4. The regular price for a video game system is $ 'O'.1t is,:>:1 sale for 20% off. VV~'1atis The amount of the discount?,#113." is the sale price? 5. Cindy earns 2. 15% commission on a.l sales. O!J. Saturday, she sold $ J,C:-:':- 0:;: merchandise. \Iy-ha was the CL-:10untof commission she earned on SarL!::-:'::'Y? 6. The band had a fundraiser and sold 25,800 worth of candy. They received 38% of :rds amount for themselves. How much did they receive?.'?age 9

10 Integers I To add integers with the same sign (both positive or both negative), add their absolute values and use the same sign. To add integers of opposite signs, find the difference of their absolute values and then take the sign of the larger absolute value. To subtract integers, add its additive inverse. For example, 6-11 = = -5 Exercises: Solve the following problems. 11. (A)',-""T I ( ~) r :» (-2)= 3.6-(-9)= 4. (-6) -7 = (-9) = 6. 1~ - 24 = 7. (-5) -!- (-8) = = (-4) - 8 = ~O. ~4.5-29= = :-(-19.5)= (-5)= '5-(-13.5.,-(-1~1= ::;4+ (-6.:S) = L ';"8.::;=...:... "J(),-,.;/ /0':;0 - (9'))'::' T _'- Algebra 1 Page 10 Summer 2006

11 The rules for multiplying integers are: Positive' Positive = Positive Positive' Negative = Negative Integers II Negative Negative = Positive Negative' Positive =. Jegative The rules for dividing integers are The same as multiplying integers Exercises: Solve the following problems. i-. a... (-,:.)...,. 0'" = " L. )-(''"'' - u) (A) --+ = 3. (4)(-2)(-3) = 4. (-5)( -6) -2 6(-4) (-5 - (-6)) = 8. g (-4-6) = o -6 (9- E) = j4 --~7-2 ' _ i) Co (_:; J 3..L / = /, '7 -r.., 1..J (-8) = L - (-6) -7+ = (5 -(-3)) = ~ 1,... /. ~) ( -, -) 1/. "0 - \-l.j - /. ) = ~ (-6) =, 9. (_2)3 (-5-(J)) = (-9-17) -'-:'4 = Algebra I Page 11

12 Solving Equations I The key in equation solving is to isolate the variable, to get the letter by itself. In one-step equations, we merely undo the operation - addition is the opposite of subtraction and multiplication is the opposite of division. Remember the golden rule of equation solving: If we do something to one side of the equation, we must do the exact same thing to the other side. Examples: 1. x+5== x=1 Check: = 6 6=6 3. 4x= Check: 4 (4) = = t - 6 = t = 13 Check: 13-6=7 7= e ~ = 12 c 6 r = 72 Check: 72 -i- 6 = = 12 Exercises: Solve the following problems: No Calculators! SHOW A...LLWORK Use a separate sheet of paper (if necessary) and staple to ills page. 1. x-8= t-(-9)=4. 5. y - 4 = h + 8 =-5 :? 7. -=-= :1:= p=17 Algebra I Page 12 SUIT_mer 2006

13 Solving Equations II The key in equation solving is to isolate the variable, to get the letter by itself. In two-step equations, we must undo addition and subtraction first, then multiplication and division. Remember the golden rule of equation solving: If we do something to one side of the equation, we must do the exact same tiling to the other side. Examples: x 1.4x-6= =-8-6, /" +6, x =-2 Solve: 4(-2)-6= =-14-14=-14 -v ~--.d. /" -, -0 x ~ -6 ~.d. L, Solve: (24/-6)- 4 = = -8-8 =-8 Exercises: Solve the following problems: _io Calculators! SHOW.ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page t - 6 = 22.'r. 2. -_ - 5 =-4 -) v A. -_ +(-7) = 6 -.J 5g, (-:3) = -12 Algebra 1 Page 13 Summer _006

14 Solving Equations HI \.Jihen solving equations that include basic mathematical operations, we must simplify the mathematics first, then solve the equations. For example: 5 (4-3) + 7x = 4 (9-6) 5 (1) + 7x = 4 (3) Check: 5 (4-3) + 7 (1) = 4 (9-6) 5 + 7x = = 4 (3) x= 1 12 = x=l Exercises: S lve the following equations using "[berules listed on the previous pages: SHOW ALL \VOR..1Z. Use a separate sheet or paper (if necessary) and staple to this page. ;.'"..L "(0 ')\ ~ r-: I / -....) t ~ ~1 = 8(6-4) ) 3. 5 (t - 4) = 9 (7-3) 6t = 8 (7 -.1.) t 6. 7(6-(-8'lJ =-+2, '... _.1. I. 7 (3-6) = 6 (4 - c) 8.4r--!.-:r-6r=:5-L6 ~':Jgebia 1 Page ll!. Summer 2006

15 As \,iyeknow, the key in equation solving is to isolate the variable. In equations with variables on each side of the equation, v.;e must combine the variables first by adding or subtracting the amount of one variable on each side of the equation to have a variable term on one side of the equation. Then, we must undo the addition and subtraction, then multiplication and division. Remember the golden rule of equation solving. Examples: Sx - 6 = 4x t = t - 4x - 4x + 6t + 6t 4x - 6 = 5 5 = 24 + lot x 11 lot xercises: Solve the following problems: No Calculators! SHOW ALL \VORK. Use a separate sheet 0:paper (if necessary) and staple to this page. l. Lir - 7 = Sr t = 5t x...L 5 = 3x y.-5=4y x-8==6-::): 6. ';'p - 8 = _Lip.J.. 6.' Algebra l Page 15 SClIT'JDer 2006

16 Ineq ualities In solving inequalities, the solution process is very similar to solving equalities. The goal is still to isolate the variable, to get the letter by itself. However, the one difference between equations and inequalities is that when solving inequalities, when we multiply or divide by a negative number, we must change The direction of the inequality. Also, since an inequality as many solutions, we can represent the solution of an inequality by a set of numbers or by the numbers on a number line. Ineouality - a statement containing one of the following symbols: < is less than > is greater than < is less than or equal to ::::is greater than or equal to is not equal to Exarncles: 1. Integers between -4 and 4..., -,) -2 w -1.""' ~-~~~ o 2. All numbers between -4 and 4. -:: The positive numbers. < 1! , I ).L... '* 3 son, therefore -c- A > 2 is L:!.e solutio (this is because whenever we multiply 0:- c.vide an inequality by a neganve number, the direction or the inequality must change) and can be represented as: ~~- -: - s: I A... 1! s;:e:-c:ses: Solve the fol;o-,a':':r:gproblems: Calculators! 1. 4x.> 9.-_ ~,.., <I I I 1 1! I.,..!.:::z 53 \.. _ T g; r- -1' f.i I I 1 1 -,..., I"" :i B 7 -. A -( -::> -05 ::;. I I b 9 I 0 -St > -15 _-I I I I~ '--!'j,, 1 1 "-" - I * 5 1 1! 1 I I I 1 I I J - ~,. -,.,..J. - 2:.J ').., , I, I ::1 -':: :'J I I.-, I , 1 1 I I~ I I 1 I J r-' d ::;., <oj X 4. -> 2-4 -I 1 _J. _ J ~i *" 1 J ~ 1~_ I """ I I... -;; :> ( 1 1, I J, I I~ 1 :'~ J Algebra 1 Page 16 Summer 2006

17 Pythagorean Theorem The Pythagorean Theorem states that in a right tri3.i.1qje, and only in a right triangle, the length of the longest side (the side opposite the right angle and called the hypotenuse, or c in the formula) squared is equal to the sum of the squares of the other two sides (the sides that meet to form the. ht 1.. d l d l t, r- ng t ang e calle ~,or a an 0 In the rorrnu I) a. The 1 e f rormula..? IS a- + b? - = C-.? Find the missing side. 7~?. 'j 2 a- + b- = c x 2 = x 2 = x 2 = 576./~2 _ /)-/~ \.A.. - '\! : 0 x =24 Exercises: Solve for the variable: SHO'lI.A~L VJOP K. Use a separate sheet of paper (if necessary) 2:1d stapie to this page. 1. 4ft u 3m 5D y 4ft x ~\.., \.i ~. \ \\ 17cm -c-.'- \ \ I \\ n \ 8c:n 12 ft u I - r- ):I I 5. I :24211,! L I, \. \ \ 6. '7 ;. 10m 10m 16ft~ Algebra 1 Page 17

18 t..::.ints/guide: Volume To find the volume of prisms (a solid figure whose ends are parallel and the same size and shape and whose sides 3Ie parallelograms) and cylinders, we multiply the area of the base times the height of the figure. The formulas we need to know are: The area of a circle is A = 1'[ r2 1 The area of a triangle is A = - b h 2 The area of a rectangle is A = bh The volume of a prism is v = (Area of Base) e (Height) So, the volume of a rectangular prism can be determined if we can find the area of the base and the perpendicular height of the figure. Exercises: Findthevollliueofthefollowingfig1L"es: Note: Use Ii: =3.14 SHOW ALL WO~T(. Use a separate sheet of paper (if necessary) and staple to this page LJ ~... // ~/ I /1 1 /.).'. / ~m // l~m 8m /-- ( r... I ~~-- I ~.J. A -f. //r~,. I I I 10 in I 18 ~ I /'0... i l '/9 'it' "".,.~ (I!:.. o Lt. TI.. ". / n "'" 8ft Page 18 Summer 2006

19 To determine the surface area of an object, vse must find the areas of each surface and add them together. For a rectangular prism, we find the area of each rectangle and then add them together. For a cylinder, we find the area of each base and then add the area of the rectangle (the circumference of the circular base times the height) which wraps around to create the sides of the cylinder. For example: The area of each base is A = m 2 = = m 2 and the area of the cylinder "wrap" is A = 2mh (which is the circumference of the circle times the height of the cylinder) = $ 12 = So the surface area is = m".=xe:-c:ses: Determine the surface area of.he following figures: =~O-:v ALL -T~-OR... K. Use a separate sheet of paper (if necessary) and staple to this page /1,/1 / : /' I /.....,T..... '.' [7... ' ~--f,//.' I /. ~... I,/.z m... 1'// t" v" 2~ in f---//'------'1i 4. //1'" / "-./ '" /' I -, r r j i i! I. 8 r- I 1 ::1..',. 10 in 8ft Algebra 1?age 19 Summer 2006

20 Exercises: Solve each problem. SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page. 1. The Acme Supply Store sells a security system for $ excluding tax. They sold 12 systems. If the total profit on these sales was $ , how much did each system cost Acme Supply? Show your work. 2. Kristen is paid $5.60 per hour. She works 6 hours on Saturday, 3 hours on Sunday, and 1 5 hours on Monday. On Sz.urday her hourly rate in 1- times her regular rate and, 2 she is paid twice the regular rate on Sunday. How much did she e2.d in all? Show all work. 3. A-the. beginning of The week the value of a s:ock 'V\'2.S32~~. On Monday it fell /:, on Tuesday it rose 1~i,on Wednesday it rose 3, 0::1 Thursday it fell 2, and 0:1 Friday it,ost 212. What '\1,'25 the value a: the stock at the end of the week? Show 2.1i work. :i Norma is paid $4.80 per hour. She worked 3 ~ ihours on Friday, 4 hours on Saturday, acid 2Y:zhours on Sunday. On Saturday, her hourly rate was 1~I;times her regular ;Jay and on Sunday, it 'v, 2.S twice the regular rate. How much did she earn in 2.1~? Show all work. ). hie formula which converts Fahreahe.t degrees (F) to Celsius cegrees (C) is: 5 C = - )" - 32). How r:13...iy c.eg::-eescelsius is l ~3 '?? 9 6. The formula for the.perimeter of a rectangle is P = 2(1 + 'v,;). If the length is 8 inches a:lc the width is 7 inches, determine the perimeter of the rectangle. Page 20

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