1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

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1 ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look for mistkes. If you hve troule, sk mth techer or someone else who understnds this topic. TOPIC : ARITHMETIC OPERATIONS A. Frctions: Simplifying frctions: exmple: Reduce 7 6 : (Note tht you must e le to find common fctor; in this cse 9; in oth the top nd the ottom in order to reduce.) Prolems -: Reduce: Equivlent frctions: exmple: is equivlent to how mny eighths? Prolems -5: Complete: How to get the lowest common denomintor (LCD) y finding the lest common multiple (LCM) of ll denomintors: exmple: 5 6 nd 8. First find LCM of 6 nd 5: LCM 5 0, so 5 6 0, nd Prolems 6-7: Find equivlent frctions with the LCD: 6. nd nd 7 8. Which is lrger, 5 7 or? (Hint: find the LCD frctions) Adding, sutrcting frctions: if the denomintors re the sme comine the numertors: exmple: Prolems 9-: Find the sum or difference (reduce if possile): If the denomintors re different, find equivlent frctions with common denomintors, then proceed s efore: exmple: exmple: Multiplying frctions: multiply the top numers, multiply the ottom numers, reduce if possile. exmple: ( ) Dividing frctions: mke compound frction, then multiply the top nd ottom (of the ig frction) y the LCD of oth: exmple: exmple: B. Decimls: Mening of plces: in.59, ech digit position hs vlue ten times the plce to its right. The prt to the left of the point is the whole numer prt. Right of the point, the plces hve vlues: tenths, hundredths, etc.,.59 ( 00)+ ( 0)+ ( ) So, + ( 5 0)+ ( 00)+ ( 9 000)..Which is lrger:.59 or.7? To dd or sutrct decimls, like plces must e comined (line up the points). exmple:... exmple: +.. exmple: 6.0 (.)

2 $.5 $.68 Multiplying decimls: exmple:..5.5 exmple:...06 exmple: (.0) (.5) Dividing decimls: chnge the prolem to n equivlent whole numer prolem y multiplying oth y the sme power of ten. exmple:..0 Multiply oth y 00, to get 0 0 exmple:.0 Multiply oth y 000, get C. Positive integer exponents nd squre roots of perfect squres: Mening of exponents (powers): exmple: 8 exmple: 6 Prolems 5-: Find the vlue: (.). (.) is non-negtive rel numer if 0 mens, where 0. Thus 9 7, ecuse 7 9. Also, D. Frction-deciml conversion: Frction to deciml: divide the top y the ottom. exmple:.75 exmple: exmple: Prolems 5-55: Write ech s deciml. If the deciml repets, show the repeting lock of digits: Non-repeting decimls to frctions: red the numer s frction, write it s frction, reduce if possile: exmple:. four tenths 0 5 exmple:.76 three nd seventy six hundredths Prolems 56-58: Write s frction: E. Percents: Mening of percent: trnslte percent s hundredths : exmple: 8% mens 8 hundredths or.08 or To chnge deciml to percent form, multiply y 00: move the point plces right nd write the % symol. exmple: % exmple:.5 5% Prolems 59-60: Write s percent: To chnge percent to deciml form, move the point plces left nd drop the % symol. exmple: 8.76%.0876 exmple: 67%.67 Prolems 6-6: Write s deciml: 6. 0% 6..0% To solve percent prolem which cn e written in this form: % of is c First identify,,c : Prolems 6-65: If ech sttement were written (with the sme mening) in the form of % of is c, identify,, nd c : 6. % of 0 is is 50% of out of is 5% Given nd, chnge % to deciml form nd multiply (since of cn e trnslted multiply ). Given c nd one of the others, divide c y the other (first chnge percent to deciml, or if the nswer is, write it s percent).

3 exmple: Wht is 9.% of $5000? ( % of is c : 9.% of $5000 is? ) 9.% $5000 $70 (nswer) exmple: 56 prolems correct out of 80 is wht percent? ( % of is c :? % of 80 is 56) % (nswer) exmple: 560 people vote in n election, which is 60% of the registered voters. How mny re registered? ( % of is c : 60 % of? is 560); 60%.6; (nswer) 66. % of 9 is wht? 67. Wht percent of 70 is 56? 68. 5% of wht is 60? F. Estimtion nd pproximtion: Rounding to one significnt digit: exmple:.67 rounds to exmple:.09 rounds to.0 exmple: 850 rounds to either 800 or 900 Answers: , , 8. (ecuse 0 8 < 8 ) TOPIC : POLYNOMIALS $ not rel numer Prolems 69-7: Round to one significnt digit To estimte n nswer, it is often sufficient to round ech given numer to one significnt digit, then compute. exmple: Round nd compute: is the estimte Prolems 7-75: Select the est pproximtion of the nswer: (, 0, 00, 000, 0000) (.0,.,.5, 5, 0, 50) (,, 98, 05, 00) 75. (.68590) (, 6, 60, 5000, 000) % % c % A. Grouping to simplify polynomils: The distriutive property sys: ( + c) + c exmple: ( x y ) x y (, x, c y)

4 x + 7x ( + 7)x x exmple: ( x,, c 7) exmple: + 6x +x Prolems -: Rewrite, using the distriutive property:. 6( x ). 5( ). x x Commuttive nd ssocitive properties re lso used in regrouping: exmple: x + 7 x x x + 7 x + 7 exmple: 5 x x 0 x exmple: x + y x + y x x + y + y x + 5y Prolems -9: Simplify:. x + x 7. x ++ x x x y + y 8x 9. x y + y x B. Evlution y sustitution: exmple: If x, then 7 x 7 () 7 5 exmple: If 7 nd, then ( 7) ( ) 9( ) 9 exmple: If x, then x x 5 ( ) ( ) Prolems 0-9: Given x, y, nd z. Find the vlue: 0. x 5. x + y. z 6. x x. xz 7. ( x + z). y + z 8. x + z. y + z 9. x z C. Adding, sutrcting polynomils: Comine like terms: exmple: ( x + x +) ( x ) x + x + x + x + exmple: ( x ) + ( x + x ) x + x + x x + x exmple: ( x + x ) ( 6x x +) x + x 6x + x 5x + x Prolems 0-5: Simplify: x + 0. x + x. ( x )+ ( 5 x). ( )+ ( + ). ( y y 5) ( y y + 5). ( 7 x) ( x 7) 5. x ( x + x ) D. Monomil times polynomil: Use the distriutive property: exmple: ( x ) x + ( ) x + ( ) x exmple: ( x + ) x + exmple: x( x ) x + x 0. ( x ) 6. x 7 7. ( ). ( x ) ( ) 8. x( x + 5). 8( + 7) 9. ( x )7 E. Multiplying polynomils: Use the distriutive property: ( + c) + c ( x ) is + c exmple: x + if: ( x +), x, nd c So ( + c) + c ( x +)x + ( x +) ( ) x + x 8x x 7x Short cut to multiply ove two inomils (see exmple ove): FOIL (do mentlly nd write the nswer) F: First times First: ( x) ( x) x O: multiply Outers : ( x) ( ) 8x I: multiply Inners : ( ) ( x) x L: Lst times Lst: ( ) ( ) Add, get x 7x exmple: ( x + ) ( x + ) x + 5x + 6 exmple: ( x ) ( x + ) x + x exmple: ( x 5) ( x + 5) x 5 exmple: ( x ) x + exmple: ( x ) 9x x +6 exmple: ( x + ) ( 5) x 5x + 5 Prolems -: Multiply:. ( x + ) 8. 6x( x). ( x ) 9. ( x ) 5. ( x + ) ( x ) 0. ( x ) ( x + ) 6. ( x + ) ( x ). ( x ) ( x + ) 7. ( x ) ( x )

5 F. Specil products: These product ptterns (exmples of FOIL) should e rememered nd recognized: I. ( + ) ( ) II. ( + ) + + III. + Prolems -: Mtch ech pttern with its exmple:. ( x ) 9x 6x +. ( x + 5) x + 0x + 5 c. ( x + 8) ( x 8) x 6. I:. II:. III: Prolems 5-5: Write the nswer using the pproprite product pttern: 5. ( +) ( ) 9. ( ) ( ) 50. ( x y) 5. ( x +y) ( ) 5. ( x + y) x y 6. y G. Fctoring: Monomil fctors: + c ( + c) exmple: x x x( x ) 5 exmple: x y + 6xy xy( x + ) Difference of two squres: + ( ) ( x ) exmple: 9x x + Trinomil squre: exmple: x 6x + 9 ( x ) Trinomil: exmple: x x ( x ) ( x +) exmple: 6x 7x x + Prolems 5-67: Fctor: ( x ) x x x x 55. 8x 6. 8x + 8x + x 56. x 0x x +x xy +0x 65. 6x y 9x y 58. x x x x 59. x x x 0x x y y x Answers:. 6x 8. x x x + y 7. 5x x 9. x y x. x.. y y 0. x 5. x + 6. x x + 5x 9. x 7 0. x. x x + 6x + 9. x 6x x 9 6. x 9 7. x 6x x + 6x 9. x x + 0. x + x. x + x. c y y x xy + y 5. 6x + xy + 9y 5. 9x y ( x ) ( x +) ( x + ) ( x + ) ( + x) ( x ) 55. x x x y 5x 58. x x 60. xy x y 6. x 5 6. x x 6. x x + 6. x x y y x 66. x 67. x

6 6 TOPIC : LINEAR EQUATIONS nd INEQUALITIES A. Solving one liner eqution in one vrile: Add or sutrct the sme vlue on ech side of the eqution, or multiply or divide ech side y the sme vlue, with the gol of plcing the vrile lone on one side. If there re one or more frctions, it my e desirle to eliminte them y multiplying oth sides y the common denomintor. If the eqution is proportion, you my wish to cross-multiply. Prolems -: Solve:. x 9 7. x 6 x. 6x 5 8. x x +. x x x x x 5 x x 9. x + 5 x 6. x x + 5 To solve liner eqution for one vrile in terms of the other(s), do the sme s ove: exmple: Solve for F : C 5 F 9 Multiply y 9 : 9 C F 5 5 Add : 9 C + F 5 Thus, F 9 C + 5 exmple: Solve for : + 90 Sutrct : 90 exmple: Solve for x : x + c Sutrct : x c Divide y : x c Prolems -9: Solve for the indicted vrile in terms of the other(s): y x x y x + x. P + h 8. x + y 0 x 5. y x 9. y x 0 x y B. Solution of one-vrile eqution reducile to liner eqution: Some equtions which do not pper to e liner cn e solved y using relted liner eqution: exmple: x+ x Multiply y x : x + x Solve: x x (Be sure to check nswer in the originl eqution.) exmple: x+ 5 x+ Think of 5 s 5 nd cross-multiply: 5x + 5 x + x x But x does not mke the originl eqution true (thus it does not check), so there is no solution. Prolems 0-5: Solve nd check: 0. x x x+ x. x x+ 5. x x+8. x x+ 5. x x exmple: x Since the solute vlue of oth nd is, x cn e either or. Write these two equtions nd solve ech: x or x x x 5 x or x 5 Prolems 6-0: Solve: 6. x 9. x 0 7. x 0. x + 8. x C. Solution of liner inequlities: Rules for inequlities: If >, then: + c > + c c > c c > c (if c > 0 ) c < c (if c < 0 ) c > c (if c > 0 ) c < c (if c < 0 ) If <, then: + c < + c c < c c < c (if c > 0 ) c > c (if c < 0 ) c < c (if c > 0 ) c > c (if c < 0 ) exmple: One vrile grph: solve nd grph on numer line: x 7 (This is n revition for { x : x 7}) Sutrct, get x 6 Divide y, x Grph: Prolems -8: Solve nd grph on numer line:. x >. x <

7 . x x > x. < x 7. x > + 5. x < x + 5 x D. Solving pir of liner equtions in two vriles: The solution consists of n ordered pir, n infinite numer of ordered pirs, or no solution. Prolems 9-6: Solve for the common solution(s) y sustitution or liner comintions: 7 9. x + y 7. x y 5 x y 8 x + 5y 0. x + y 5. x y x y x + y. x y 9 5. x + y x 8 x + y. x y 6. x y y x 5 6x 9 y Answers: ( P h). ( y +) y ( y ) y 9. x y TOPIC : QUADRATIC EQUATIONS no solution 6., 7. no solution 8., 9. 0.,. x > 7. x < x 5 0. x > x > 6. x < 7. x > x - 9. ( 9, ) 0. (, ). (8, 5). (, 9). 8 ( 9, 9 ). (,0) 0 5. no solution 6. ny ordered pir of the form (, ) where is ny numer. One exmple: (, 5). Infinitely mny solutions. A. x + x + c 0: A qudrtic eqution cn lwys e written so it looks like x + x + c 0 where,, nd c re rel numers nd is not zero. exmple: 5 x x Add x: 5 x + x Sutrct 5: 0 x + x 5 or x + x 5 0 So,, c 5 exmple: x Rewrite: x 0 (Think of x + 0x 0) So, 0, c Prolems -: Write ech of the following in the form x + x + c 0 nd identify,, c :. x + x 0. x x. 5 x 0. x x 5. 8x B. Fctoring: Monomil fctors: + c + c exmple: x x x x exmple: x y + 6xy xy x + Difference of two squres: + ( ) ( x ) exmple: 9x x +

8 Trinomil squre: exmple: x 6x + 9 ( x ) Trinomil: ( x +) ( x ) exmple: x x x exmple: 6x 7x x + Prolems 6-0: Fctor: x x x x 8. 8x 6. x + 8x + 8x 9. x 0x x +x + 0. xy +0x 8. 6x y 9x y. x x 5 9. x x. x x 6 0. x 0x +. x y y x C. Solving fctored qudrtic equtions: The following sttement is the centrl principle: If 0, then 0 or 0. First, identify nd in 0: exmple: ( x) ( x + ) 0 Compre this with 0 ( x); ( x + ) Prolems -: Identify nd in ech of the following:. x( x 5) 0. ( x ) ( x 5) 0. ( x )x 0. 0 ( x ) ( x +) Then, ecuse 0 mens 0 or 0, we cn use the fctors to mke two liner equtions to solve: exmple: If x( x ) 0 then( x) 0 or x 0 nd so x 0 or x ; x. Thus, there re two solutions: 0 nd exmple: If ( x) ( x + ) 0 then x or ( x + ) 0 nd thus x or x. 0 0 exmple: If x + 7 then x x 7 x 7 (one solution) 8 Note: there must e zero on one side of the eqution to solve y the fctoring method. Prolems 5-: Solve: 5. ( x +) ( x ) 0 9. ( x 6) ( x 6) 0 6. x( x + ) 0 0. ( x ) ( x 5)x. x( x + ) x 8.0 x + ( x ) 0 D. Solving qudrtic equtions y fctoring: Arrnge the eqution so zero is on one side (in the form x + x + c 0), fctor, set ech fctor equl to zero, nd solve the resulting liner equtions. exmple: Solve: 6x x Rewrite: 6x x 0 Fctor: x x So x 0 or x 0 0 Thus, x 0 or x exmple: 0 x x 0 ( x ) ( x + ) x 0 or x + 0 x or x Prolems -: Solve y fctoring: ( x + ) ( x ) ( x ) 0. x x. x x 0 9. x +. x x 0. 6x x + 5. x( x + ) x 6x 6. x x. x x 7. x + x 6. x x 6 0 Another prolem form: if prolem is stted in this form: One of the solutions of x + x + c 0 is d, solve the eqution s ove, then verify the sttement. exmple: Prolem: One of the solutions of 0x 5x 0 is A. B. C. D. E. 5 Solve 0x 5x 0 y fctoring: 5x( x ) 0 so 5x 0 or (x ) 0 thus x 0 or x. Since x is one solution, nswer C is correct.

9 . One of the solutions of x A. B. C. 0 D. E. ( x + ) 0 is 9 5. One solution of x x 0 is A. B. C. D. E. Answers: c. x + x 0 -. x x x x x x Note: ll signs could e the opposite ( x -) ( x +) ( x + ) 8. x + 9. x 5 0. x y 5x. x 5. x. xy x y TOPIC 5: GRAPHING ( x + ) ( + x) ( x ). x 5 5. x x 6. x x + 7. x + 8. x y y x 9. x 0. x. x x 5. x x. x x 5. x x + 5., 6., , 5 8., , 0,. 0,. 0,. 0, 5., 0 6., 7., 8., 9., 0.,..,.,. B 5. B A. Grphing point on the numer line: Prolems -7: Select the letter of the point on the numer line with coordinte: A B C D E F G Prolems 8-0: Which letter est loctes the given numer: Prolems -: Solve ech eqution nd grph the solution on the numer line: exmple: x + x. x 6 0. x + x. x x + 5 B. Grphing liner inequlity (in one vrile) on the numer line: Rules for inequlities: If >, then: + c > + c c > c c > c (if c > 0) c < c (if c < 0) (if c > 0) > c c < c c (if c < 0) If <, then: + c < + c c < c c < c (if c > 0) c > c (if c < 0) (if c > 0) < c c > c c (if c < 0) exmple: One vrile grph: solve nd grph on numer line: x 7 (This is n revition for { x : x 7}) Sutrct, get x 6 Divide y -, x Grph:

10 0 Prolems -0: Solve nd grph on numer line:. x > 8. x < 6 5. x < 9. 5 x > x 6. x x >+ 7. < x exmple: x > nd x < The two numers - nd splits the numer line into three prts: x >, < x <, nd x <. Check ech prt to see if oth x > nd x < re true: prt x vlues x >? x <? oth true? x < < x < no yes yes yes no yes (solution) x > yes no no Thus the solution is < x < nd the grph is: exmple: x or x < ( or mens nd/or ) prt x vlues x? x <? x x < x > yes no no yes yes no t lest one true? yes (solution) yes (solution) no So x or x <; these cses re oth covered if x <. Thus the solution is x < nd the grph is: Prolems -: Solve nd grph:. x < or x >. x 0 nd x >. x > nd x C. Grphing point in the coordinte plne: If two numer lines intersect t right ngles so tht: ) one is horizontl with positive to the right nd negtive to the left, ) the other is verticl with positive up nd negtive down, nd ) the zero points coincide Then they form coordinte plne, nd ) the horizontl numer line is clled the x-xis, ) the verticl line is the y-xis, ) the common zero point is the origin, ) there re four y qudrnts, II I numered x s shown: III IV To locte point on the plne, n ordered pir of numers is used, written in the form (x, y). The x-coordinte is lwys given first. Prolems -7: Identify x nd y in ech ordered pir:. (, 0) 6. (5, -) 5. (-, 5) 7. (0, ) To plot point, strt t the origin nd mke the moves, first in the x-direction (horizontl) nd the in the y-direction (verticl) indicted y the ordered pir. exmple: (-, ) Strt t the origin, move left (since x ), then (from there), up (since y ) Put dot there to indicte the point (-, ) 8. Join the following points in the given order: (-, -), (, -), (-, 0), (, ), (-, ), (, 0), (-, -), (-, ), (, -) 9. Two of the lines you drew cross ech other. Wht re the coordintes of this crossing point? 0. In wht qudrnt does the point (, ) lie, if > 0 nd < 0? Prolems -: For ech given point, which of its coordintes, x or y, is lrger? D. Grphing liner equtions on the coordinte plne: The grph of liner eqution is line, nd one wy to find the line is to join points of the line. Two points determine line, ut three re often plotted on grph to e sure they re colliner (ll in line). Cse I: If the eqution looks like x, then there is no restriction on y, so y cn e ny numer. Pick numers for vlues of y, nd mke ordered pirs so ech hs x. Plot nd join.

11 exmple: x Select three y s, sy -, 0, nd Ordered pirs: (-, -), (-, 0), (-, ) Plot nd join: Note the slope formul 0 gives ( ) 0, which is not defined: verticl line hs no slope. Cse II: If the eqution looks like y mx +, where either m or (or oth) cn e zero, select ny three numers for vlues of x, nd find the corresponding y vlues. Grph (plot) these ordered pirs nd join. exmple: y Select three x s, sy -, 0, nd Since y must e -, the pirs re: (-, -), (0, -), (, -) The slope is ( ) And the line is horizontl. exmple: y x Select x s, sy 0,, : If x 0, y 0 If x, y If x, y 5 Ordered pirs: (0, -), (, ), (, 5) Note the slope is ( ) 0, And the line is neither horizontl nor verticl. Prolems 5-: Grph ech line on the numer plne nd find its slope (refer to section E elow if necessry): 5. y x 9. x 6. x y 0. y x 7. x y 8. y. y x + E. Slope of line through two points: Prolems -7: Find the vlue of ech of the following: ( ) ( ) nd The line joining the points P x, y hs slope y y P x, y x x. exmple: A (, -), B (-, ) slope of AB ( ) 5 5 Prolems 8-5: Find the slope of the line joining the given points: 8. (-, ) nd (-, -) 9. (0, ) nd (-, -5) 50. (, -) nd (5, -) Answers:. D. E. C. F 5. B 6. G 7. B 8. Q 9. T 0. S.. 5. ½. x > 7 5. x < 6. x ½ 0 7. x > 6 8. x > 9. x < 0. x > 5. x < or x >. x >. < x x y (0,-) 0. IV. x. y. y. x none - -

12 0. -. ½ -. ½ none (undefined) TOPIC 6: RATIONAL EXPRESSIONS A. Simplifying frctionl expressions: exmple: (note tht you must e le to find common fctor in this cse 9 in oth the top nd ottom in order to reduce frction.) exmple: (common fctor: ) Prolems -: Reduce: c x x ( x+) ( x 5) ( x 5) ( x ). 6xy 5y 0. x 9x x 9. 8( x ) 6 x x 7y x. x 7y x x+ exmple: y 0x 0 x y x 5 y 5 x y 5 5 x x y y y y y Prolems -: Simplify: x. xy y 6 y. x x x B. Evlution of frctions: exmple: If nd, find the vlue of + Sustitute: + () x( x ) x 6 Prolems 5-: Find the vlue, given,, c 0, x, y, z : x 5y y x x c x 7.. z y 8. c. z C. Equivlent frctions: exmple: is equivlent to how mny eighths? 8, 6 8 exmple: exmple: x+ x+ ( x+) x+ x+ x+8 x+ x+ x+ exmple: x x+ ( x+) ( x ) x ( x ) ( x ) x+ ( x ) ( x+) x x+ ( x+) ( x ) Prolems -7: Complete: ( +) ( ). x 7 7y 7. x 6 6 x x+ 5. x+ ( x ) ( x+) How to get the lowest common denomintor (LCD) y finding the lest common multiple (LCM) of ll denomintors: exmple: 5 6 nd 8 5. First find LCM of 6 nd 5: LCM 5 0, so , nd exmple: nd 6 : 6 LCM, so 9, nd 6 exmple: x+ nd x LCM ( x + ) ( x ), so x+ x ( x+) ( x ), nd x x+ ( x+) ( x ) Prolems 8-: Find equivlent frctions with the lowest common denomintor: 8. nd 9. x nd x 9. x nd 5. x nd 5 x+ 0. x nd x+. x nd x x+

13 D. Adding nd sutrcting frctions: If denomintors re the sme, comine the numers: exmple: x x x x x y y y y Prolems -8: Find the sum or difference s indicted (reduce if possile): x+ x +x y xy 5. x x x If denomintors re different, find equivlent frctions with common denomintors, then proceed s efore (comine numertors): exmple: exmple: + x x+ ( x+) ( x ) ( x+) + ( x ) ( x ) ( x+) ( x+6+x x ) ( x+) x+5 ( x ) ( x+) Prolems 9-5: Find the sum or difference: x 7. x + x x x. 5 x 8. x x x x x x+ x. 50. x x x x. c 5. x x x 5. + E. Multiplying frctions: Multiply the tops, multiply the ottoms, reduce if possile: exmple: exmple: ( x+) x x x ( x+) ( x+) ( x ) ( x ) ( x+) ( x ) x+6 x ( ) 5. c d ( x+) x+ 5y 5y x 6 x x x+6 F. Dividing frctions: A nice wy to do this is to mke compound frction nd then multiply the top nd ottom (of the ig frction) y the LCD of oth: exmple: c d c d 7 exmple: d d c c d d x 5x exmple: 5x x y y x y y x y 5x x 9 x+7 x c c 5 xy y Answers:.. 5. x x y y c x ( x +) x. ( x ) ( x +). x+ x. x. x undefined xy ( x + ) ( ) 5. x + x or x 6. + or , , 5x x x 0. x ( x + ) ( x +), ( x +)., x x

14 . ( x+) ( x ) ( x+), 5( x ) ( x ) ( x +), x x( x +). x + x x x x x. x 0 5x. 5.. c x +x 9. x ( x ) ( x+) ( x ) 50. x + x 5. x +x x c d y x 59. x x +7 x c 7. c TOPIC 7: EXPONENTS nd SQUARE ROOT A. Positive integer exponents: mens use s fctor times. ( is the exponent or power of.) exmple: 5 mens, nd hs vlue. exmple: c c c c Prolems -: Find the vlue:. 8. (.) ( ) exmple: Simplify: 5 Prolems 5-8: Simplify: 5. x 7. x y B. Integer exponents: I. c +c II. c III. IV. V. c ( ) c c ( ) c c c c c c ( x) x VI. 0 (if 0) VII. Prolems 9-8: Find x: 9. x. 8 x 0. x 5. x x x x 7. c. x 8. y y Prolems 9-: Find the vlue: c x 9. 7x 0 x 6. c+ x c 0. x 7. 6x. 8. x x x x x. ( ). xy 5. x c+ x c C. Scientific nottion: exmple: if the zeros in the ten s nd one s plces re significnt. If the one s zero is not, write.80 0 ; if neither is significnt:.8 0. exmple:

15 exmple: 0 00 exmple: Note tht scientific form lwys looks like 0 n where <0, nd n is n integer power of 0. Prolems -5: Write in scientific nottion:. 9,000, Prolems 6-8: Write in stndrd nottion: To compute with numers written in scientific form, seprte the prts, compute, then recomine. exmple: (. 0 5 ) (.) exmple: exmple: Prolems 9-56: Write nswer in scientific nottion: D. Simplifiction of squre roots: (.9 0 )(. 0 7 ) 8. 0 if nd re oth non-negtive ( 0 nd 0). exmple: 6 exmple: exmple: If x 0, x 6 x Note: If x < 0, x 6 x mens (y definition) tht ), nd ) 0 Prolems 57-69: Simplify (ssume ll squre roots re rel numers): x x E. Adding nd sutrcting squre roots: exmple: exmple: Prolems 70-7: Simplify: F. Multiplying squre roots: if 0 nd 0. exmple: 6 6 exmple: 6 exmple: ( 5 ) ( )5 5 0 Prolems 7-79: Simplify: ( ) ( ) 79. Prolems 80-8: Find the vlue of x: x 8. 5 x G. Dividing squre roots:, if 0 nd > 0. exmple: (or 8 ) Prolems 8-86: Simplify: If frction hs squre root on the ottom, it is sometimes desirle to find n equivlent frction with no root on the ottom. This is clled rtionlizing the denomintor. 5 exmple: exmple: 9

16 Prolems 87-9: Simplify: Answers: x x 8. y y x c 6. x 6 7. x 8. x 9 9. x x 6. 8x y x x 67. x TOPIC 8: GEOMETRIC MEASUREMENT A. Intersecting lines nd prllels: If two lines intersect s shown, djcent ngles dd to 80. For exmple, c d + d 80. Non-djcent ngles re equl: for exmple, c. If two lines, nd, re prllel nd re cut y third line c, c forming ngles w, x, y, z w x y s shown, then z x z, w + y 80, so z + y 80. exmple: If x nd c x, find the mesure of c. c, so x. + 80, so x + x 80, giving x 80, or x 5 Thus c x 5 Prolems -: Given x 7, find the mesures of the other ngles:. t. z. y. w c t x y z w

17 5. Find x: x B. Formuls for perimeter P nd re A of tringles, squres, rectngles, nd prllelogrms: Rectngle, se, ltitude (height) h: P + h A h If wire is ent in the shpe, the perimeter is the length of the wire, nd the re is the numer of squre units enclosed y the wire. exmple: Rectngle with 7 nd h 8: P + h units A h sq. units A squre is rectngle with ll sides equl, so the formuls re the sme (nd simpler if the side length is s): P s s A s exmple: Squre with side cm hs P s cm A s cm (sq. cm) A prllelogrm with se nd height h hs A h. If the other side length is, then P +. exmple: Prllelogrm hs sides nd 6, nd 5 is the length of the ltitude perpendiculr to the side. P units A h 5 0 sq. units In tringle with side lengths,, c nd h is the ltitude to side, P + + c A h h exmple: P + + c units A h 0.8 sq. units x Prolems 6-: Find P nd A for ech of the following figures: 6. Rectngle with sides 5 nd Rectngle, sides.5 nd. 8. Squre with side mi. 9. Squre, side yd. s Prllelogrm with sides 6 nd, nd height 0 (on side 6).. Prllelogrm, ll sides, ltitude 6.. Tringle with sides 5,,, nd 5 is the height on side.. The tringle shown: 5 C. Formuls for circle re A nd circumference C: A circle with rdius r (nd dimeter d r) hs distnce round (circumference) C πd or C πr (If piece of wire is ent into circulr shpe, the circumference is the length of wire.) exmple: A circle with rdius r 70 hs d r 70 nd exct circumference C πr π 70 0π units. If π is pproximted y 7, C 0π 0( 7 ) 0 units pproximtely. If π is pproximted y., the pproximte C 0(.) units. The re of circle is A πr : exmple: If r 8 A πr π 8 6π sq. units Prolems -6: Find C nd A for ech circle:. r 5 units 6. d km 5. r 0 feet D. Formuls for volume V: A rectngulr solid (ox) with length l, width w, nd height h, hs volume V lwh. exmple: A ox with dimensions, 7, nd hs wht volume? V lwh 7 cu. units A cue is ox with ll edges equl. If the edge is e the volume V e exmple: A cue hs edge cm. V e 6cm (cu. cm) A (right circulr) cylinder with rdius r nd ltitude h hs V πr h 5 exmple: A cylinder hs r 0 nd h. r

18 The exct volume is V πr h π 0 00π cu. units If π is pproximted y 7, V cu. units 7 If π is pproximted y., V 00(.) 96 cu. units A sphere (ll) with rdius r hs volume V πr exmple: The exct volume of sphere with rdius 6 in. is V πr π 6 π 6 88π in Prolems 7-: Find the exct volume of ech of the following solids: 7. Box, 6 y 8 y Box, y 5 6 y Cue with edge Cue, edge.5.. Cylinder with r 5, h 0. Cylinder, r, h. Sphere with rdius r.. Sphere with rdius r. E. Sum of the interior ngles of tringle: The three ngles of ny tringle dd to 80. exmple: Find the mesures of ngles C nd A: C (ngle C) is A mrked to show its mesure is 90. B + C , so A Prolems 5-9: Given two ngles of tringle, find the mesure of the third ngle: 5. 0, , , , 7. 90, 7 F. Isosceles tringles: An isosceles tringle is defined to hve t lest two sides with equl mesure. The equl sides my e mrked: Or the mesures my e given: 7 6 Prolems 0-5: Is the tringle isosceles? 0. Sides,, 5. Sides 7,, 7 B r C 8. Sides 8, 8, The ngles which re opposite the equl sides lso hve equl mesures (nd ll three ngles dd to 80 ). exmple: Find the mesures of A nd C, C B 7 given B 65 : A + B + C 80, nd 7 A B 65, so C 50 A 6. Find mesures of A nd B, if C Find mesures of B nd C, if A Find mesure of A. 9. If the ngles of tringle re 0, 60, nd 90, cn it e isosceles? 0. If two ngles of tringle re 5 nd 60, cn it e isosceles? If tringle hs equl ngles, the sides opposite these ngles lso hve equl mesures. exmple: Find the mesures of B, AB nd AC, given this figure, nd C 0 : A 70 B B 70 (ecuse ll ngles dd to 80 ) Since A B, AC BC 6. AB cn e found with trig -- lter.. Cn tringle e isosceles nd hve 90?. Given D E 68 D nd DF 6. Find the mesure of F nd F length of FE : G. Similr tringles: If two ngles of one tringle re equl to two ngles of nother tringle, then the tringles re similr. exmple: ABC nd FED re similr: The pirs of 5 corresponding 6 A B sides re AB nd FE F BC nd ED, nd AC nd FD. A A B A B C 6 6 B 8 8 E C 8 C C 6 6 C 5 D E

19 . Nme two similr tringles nd list the pirs of corresponding sides. If two tringles re similr, ny two corresponding sides hve the sme rtio (frction vlue): exmple: the rtio to x, or x, is the sme s y nd c z. Thus, x y, x c z, nd y c z. Ech of these equtions is clled proportion Prolems -5: Write proportions for the two similr tringles:. 5. exmple: Find x: Write nd solve proportion: 5 x, so x 5, x 7 Prolems 6-9: Find x: the sum of the squres of the legs equls the squre of the hypotenuse. (The legs re the two shorter sides; the hypotenuse is the longest side.) If the legs hve lengths nd, nd the hypotenuse length is c, then + c (In words, In right tringle, leg squred plus leg squred equls hypotenuse squred. ) exmple: A right tringle hs hypotenuse 5 nd one leg. Find the other leg. Since leg + leg hyp, + x x 5 x x 6 Prolems 5-5: Ech line of the chrt lists two sides of right tringle. Find the length of the third side: leg leg hypotenuse Prolems 55-56: Find x: Find x nd y: H. Pythgoren theorem: In ny tringle with 90 (right) ngle, If the sum of the squres of two sides of tringle is the sme s the squre of the third side, the tringle is right tringle. exmple: Is tringle with sides 0, 9, right tringle? 0 + 9, so it is right tringle. Prolems 57-59: Is tringle right, if it hs sides: 57. 7, 8, , 6, 58., 5, 6 Answers: P A 6. 0 un 50 un 7. un 6 un 8. mi 9 mi 9. yd 9 6 yd 0. 0 un 60 un. 8 un 7 un. 0 un 0 un. un 6 un C A. 0π un 5π un 5. 0π ft 00π ft 6. π km π km

20 . 50π. 6π. π. 9π no. yes. yes. yes. yes 5. cn t tell TOPIC 9: WORD PROBLEMS ech 7. 0, no 0. no. yes:., 6. ABE, ACD AB, AC AE, AD BE, CD d c f f +e , yes 58. no 59. yes A. Arithmetic, percent, nd verge:. Wht is the numer, which when multiplied y, gives 6?. If you squre certin numer, you get 9. Wht is the numer?. Wht is the power or 6 tht gives 6?. Find % of is wht percent of 88? 6. Wht percent of 55 is 88? 7. 5 is 80% of wht numer? 8. Wht is 8.% of $7000? 9. If you get 6 on 0-question test, wht percent is this? 0. The 00 people who vote in n election re 0% of the people registered to vote. How mny re registered? Prolems -: Your wge is incresed y 0%, then the new mount is cut y 0% (of the new mount):. Will this result in wge which is higher thn, lower thn, or the sme s the originl wge?. Wht percent of the originl wge is this finl wge?. If the ove steps were reversed (0% cut followed y 0% increse), the finl wge would e wht percent of the originl wge? Prolems -6: If A is incresed y 5%, it equls B:. Which is lrger, B or the originl A? 5. B is wht percent of A? 6. A is wht percent of B? 7. Wht is the verge of 87, 6, 8, 59, nd 95? 8. If two test scores re 85 nd 60, wht minimum score on the next test would e needed for n overll verge of 80? 9. The verge height of 9 people is 68 inches. Wht is the new verge height if 78-inch person joins the group? B. Algeric sustitution nd evlution: Prolems 0-: A certin TV uses 75 wtts of power, nd opertes on 0 volts: 0. Find how mny mps of current it uses, from the reltionship: volts times mps equls wtts wtts kilowtt (kw). How mny kilowtts does the TV use?. Kw times hours kilowtt-hours (kwh). If the TV is on for six hours dy, how mny kwh of electricity re used?. If the set is on for six hours every dy of 0-dy month, how mny kwh re used for the month?. If the electric compny chrges 8 per kwh, wht mount of the month s ill is for TV power? Prolems 5-: A plne hs certin speed in still ir, where it goes 50 miles in three hours: 5. Wht is its (still ir) speed? 6. How fr does the plne go in 5 hours? 7. How fr does it go in x hours? 8. How long does it tke to fly 000 miles? 9. How long does it tke to fly y miles? 0. If the plne flies ginst 50 mph hedwind, wht is its ground speed?

21 . If the plne flies ginst hedwind of z mph, wht is its ground speed?. If it hs fuel for 7.5 hours of flying time, how fr cn it go ginst the hedwind of 50 mph?. If the plne hs fuel for t hours of flying time, how fr cn it go ginst the hedwind of z mph? C. Rtio nd proportion: Prolems -5: x is to y s is to 5:. Find y when x is Find x when y is 7. Prolems 6-7: s is proportionl to P, nd P 56 when s : 6. Find s when P. 7. Find P when s. Prolems 8-9: Given x y : 8. Write the rtio x : y s the rtio of two integers 9. If x, find y. Prolems 0-: x nd y re numers, nd two x s equl three y s: 0. Which of x or y must e lrger?. Wht is the rtio of x to y? Prolems -: Hlf of x is the sme s one-third of y:. Which of x nd y is the lrger?. Write the rtio x : y s the rtio of two integers.. How mny x s equl 0 y s? D. Prolems leding to one liner eqution: 5. 6 is three-fourths of wht numer? 6. Wht numer is of 6? 7. Wht frction of 6 is 5? 8. of 6 of of numer is. Wht is the numer? 9. Hlf the squre of numer is 8. Wht is the numer? is the squre of twice wht numer? 5. Given positive numer x. Two times positive numer y is t lest four times x. How smll cn y e? 5. Twice the squre root of hlf numer is x. Wht is the numer? Prolems 5-55: A gthering hs twice s mny women s men. W is the numer of women nd M is the numer of men: 5. Which is correct: M W or M W? 5. If there re women, how mny men re there? 55. If the totl numer of men nd women present is 5, how mny of ech re there? 56. $,000 is divided into equl shres. Bs gets four shres, nd Ben gets the one remining shre. Wht is the vlue of one shre? E. Prolems leding to two liner equtions: 57. Two science fiction coins hve vlues x nd y. Three x s nd five y s hve of 75, nd one x nd two y s hve vlue of 7. Wht is the vlue of ech? 58. In mixing x gm of % nd y gm of 8% solutions to get 0 gm of 5% solution, these equtions re used:.0x +.08y.05( 0), nd x + y 0 How mny gm of % solution re needed? F. Geometry: 59. Point x is on ech of two given intersecting lines. How mny such points x re there? 60. On the numer line, points P nd Q re two units prt. Q hs coordinte x. Wht re the possile coordintes of P? Prolems 6-6: A O C B 6. If the length of chord AB is x nd the length of CB is 6, wht is AC? 6. If AC y nd CB z, how long is AB (in terms of y nd z)? Prolems 6-6: The se of rectngle is three times the height: 6. Find the height if the se is Find the perimeter nd re. 65. In order to construct squre with n re which is 00 times the re of given squre, how long side should e used?

22 Prolems 66-67: The length of rectngle is incresed y 5% nd its width is decresed y 0%. 66. Its new re is wht percent of its old re? 67. By wht percent hs the old re incresed or decresed? 68. The length of rectngle is twice the width. If oth dimensions re incresed y cm, the resulting rectngle hs 8cm more re. Wht ws the originl width? 69. After rectngulr piece of knitted fric shrinks in length cm nd stretches in width cm, it is squre. If the originl re ws 0cm, wht is the squre re? 70. This squre is cut into two smller squres nd two non-squre rectngles s shown. Before eing cut, the lrge squre hd re +. The two smller squres hve res nd. Find the totl re of the two non-squre rectngles. Show tht the res of the prts dd up to the re of the originl squre. Answers: % 6. 60% $ % lower. 96%. sme (96%). B 5. 5% 6. 80% mps..075 kw..5 kwh..5 kwh. $ mph mi 7. 50x mi hr 9. y 50 hr mph. 50 z mph. 000 mi. ( 50 z)t mi : x. :. y. : x 5. x 5. M W men, 6 women 56. $ x :5, y : gm x, x + 6. x 6 6. y + z P 60, A times the originl side % 67. 5% decrese 68. 5cm ( + )