Summer Review Packet For Algebra 2 CP/Honors

Transcription

1 Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review during the er in Alger. However, some topics nd sic skills re so fundmentl to n Alger course tht MASTERY of these topics is expected nd required PRIOR to eginning Alger. This pcket covers these topics, which re: I. Solving equtions in one vrile. p. II. Solving inequlities in one vrile. p. III. Liner Equtions A. Grphing liner eqution. p. B. Finding the slope. p. C. Writing liner eqution. p. D. Writing liner eqution of prllel nd perpendiculr lines. p. IV. Solving sstem of liner equtions. A. Grphing method. p.5-6 B. Sustitution method. p.7 C. Elimintion method. p.8 V. Grphing sstem of liner inequlities. p.9 VI. Fctoring p.0 VII. Simplifing Rdicls p. VIII. Simplifing Expressions with Exponents p. IX. Solving Asolute Vlue Equtions nd Inequlities p. Prctice exercises egin on p. Answers to prctice exercises egin on p. We will e using Ti-8 or Ti-8 in our clsses. Note out the clcultors: The school will issue ech student who needs/wnts Ti-8 or Ti-8 grphing clcultor, just s we issue textook to ech student. If it is lost or roken, the student must p the replcement vlue of $90 for the TI-8 nd $05 for the TI-8. A student m prefer to purchse his/her own. If so, we recommend the Ti-8 Plus (Silver Edition optionl). We then highl recommend the clcultor e etched with the student s nme clerl on the front fce, nd on the ck for securit purposes. A sticker or nme written with mrker is not enough. During this summer ou re to complete this review pcket, following the directions for ech section. Although ou will hve some opportunit to get some help in clss when school egins, if ou need lot of help, ou should consider getting it efore school egins. You will e tested on this mteril shortl fter the eginning of school. The topics on computtion with nd without clcultor will require true mster 00% result, nd will e formtive up to tht point. We recommend ou do this pcket close to the onset of school. DO NOT LOSE THIS PACKET!! It will lso e ville from the school wesite vi the mth deprtment.

2 Alger Summer Review Pcket LHS

3 Alger Summer Review Pcket Solved Exmples I. Solving equtions in one vrile. Exmple : Solve x 9 Solution: Eliminte the + sutrcting from oth sides of the eqution. Eliminte the multipling its reciprocl,, on oth sides. Exmple : Solve 8 7 Solution: Use the distriutive propert nd comine to simplif the left side of the eqution. Eliminte the +7 sutrcting 7 from oth sides. Eliminte the dividing oth sides. Exmple : Solve Solution: Use the distriutive propert nd comine to simplif oth sides of the eqution. Eliminte the vrile term on the right sutrcting. Eliminte the 0 dding 0 to oth sides. Eliminte the dividing oth sides. Exmple : Solve ( n) 5n (n ) Solution: ( n) 5n (n ) n 5n n n n n n nosolutionor Ø or { } Continue s in the previous exmple. Correct steps hve resulted in flse sttement. Conclude tht the eqution hs no solution. LHS

4 Alger Summer Review Pcket Solved Exmples II. Solving inequlities in one vrile. Exmple 5: Solve x 6x Solution: x 6x 6x 6x 0x 0x x x x Sutrct 6x from oth sides of the inequlit. Sutrct from oth sides of the inequlit. Divide oth sides of the inequlit 0. Rememer to reverse the inequlit smol when multipling or dividing negtive numer. 5 Exmple 6: Solve x 8x Solution: x 8x 5 x x 5 x x x 5 x 5 5 x 6 Distriute. Add x to oth sides of the inequlit. Sutrct from oth sides of the inequlit. Multipl oth sides of the inequlit. 5 x x 6 Exmple 7: Solve x 0 5x x 8 Solution: x 0 5x x 0 5x x 8 x 0 x 9 x x 0 9 x x isllrel numers x 8 Use the distriutive propert nd comine to simplif oth sides of the inequlit. Sutrct x from oth sides of the inequlit. Correct steps hve resulted in true sttement. Conclude tht the solution is ll rel numers. LHS

5 Alger Summer Review Pcket Solved Exmples III. Liner Equtions A. Grphing liner eqution Exmple 8: Solution: Find the slope nd the -intercept of the line whose eqution is = x + 9 nd use them to grph the eqution. Divide to solve for. Now the eqution is in the fmilir form. The slope is the coefficient of the x-term,, nd the -intercept is the constnt term,. To grph the line, plot the -intercept, (0, ). Then, since the slope is, move units up(rise) nd units right (run) to locte second point. (See elow.) + run + rise III. Liner Equtions B. Finding the slope. C. Writing liner eqution. Exmple 9: ) Find the slope of the line pssing through the points (-, 5) nd (, 8). ) Given tht the -intercept is 6, write n eqution for the line. Solution: ) slope x x ) In liner eqution of the form mx, the slope is the coefficient of the x-term or liner term, nmel m. The numericl term or constnt term is the -intercept, nmel. So, strting with mx, sustitute ½ for m nd 6 for. Therefore, the eqution is x 6 LHS

6 Alger Summer Review Pcket Solved Exmples III. Liner Equtions D. Writing liner eqution of prllel nd perpendiculr lines. Exmple 0: ) Write n eqution of line in slope-intercept form prllel to x nd psses through the point (, ) ) Write n eqution of line in point-slope form perpendiculr to x nd psses through the point (, ) Solution: ) The slope since prllel lines hve the sme slope. Use the vlues for slope nd the given coordinte, nd sustitute the vlues for m, x, nd in the eqution mx to solve for. mx sutrct to solve for Therefore the eqution prllel to x nd psses through the point (, ) is: x ) The slope since perpendiculr lines hve opposite reciprocl slopes. The point-slope form is much esier to use. It requires sustituting in the vlues for the slope nd the point. The form is mx x. Sustitute m with nd the x with nd with tken from the given coordinte (, ). Therefore the eqution in point-slope form of the line perpendiculr to x nd psses through the point (, -) is: x LHS 5

7 IV. Solving sstem of liner equtions. A. Grphing method. Alger Summer Review Pcket Solved Exmples Exmple : Solve the following sstem grphing. x x Solution: To solve the sstem of equtions grphing, ou need to grph oth equtions crefull nd then locte the point of intersection. The point of intersection is the solution to the sstem. Since oth equtions re lred in slope-intercept form it is es to extrct the slope nd -intercept of ech eqution nd use tht informtion to grph ech ccordingl s shown previousl in the section on pge clled III. Liner Equtions A. Grphing liner eqution. The solution (the point of intersection) is (, ) Exmple : Solve the following sstem grphing. x 6 x Solution: Put ech eqution into slope-intercept form. The solution is (, ) LHS 6

8 Alger Summer Review Pcket IV. Solving sstem of liner equtions. A. Grphing method. Solved Exmple Exmple : Solve the following sstem grphing. x 6 7 x Solution: Put ech eqution into slop-intercept form. The -intercepts for the two equtions re not nice whole numer, thus mking it difficult to grph those points. Now mke tle of x nd vlues nd find nice points to grph. Onl two points re needed to grph line. x x Use these two points to grph Use these two points to grph The solution is (5, ) LHS 7

9 IV. Solving sstem of liner equtions. B. Sustitution method. Alger Summer Review Pcket Solved Exmple Exmple : Solve the following sstem using sustitution. x x 7 Solution: To solve the sstem of equtions sustitution, ou need to solve for x or with one of the equtions. Your choice here cn sve time. Let s choose to solve the st eqution for x. This will require onl step. Sustitute the equivlent expression for x into the other eqution. Continue to solve for ccordingl. At this point sustitute the solution for into either of the two originl equtions. Agin our choice here cn sve time. Let s choose eqution. The solution is (-5, 6) LHS 8

10 IV. Solving sstem of liner equtions. C. Elimintion method. Alger Summer Review Pcket Solved Exmple Exmple 5: Solve the following sstem using sustitution. x 5 x 9 Solution: To solve the sstem of equtions elimintion, ou need to write equivlent equtions contining the sme coefficient for either x or. Multipl the first eqution nd the second 5. Then the vrile cn e eliminted dding the two equtions. Multipl Multipl 5 Find sustituting 7 for x in either eqution. Let s use the st one. The solution is (7, 5) LHS 9

11 V. Grphing sstem of liner inequlities. Grphing Liner Inequlit Slope- intercept form: < mx + Alger Summer Review Pcket Solved Exmples line shding st grph the -intercept nd use the slope to grph one or more points. Connect points to mke solid or dshed line. Shde ove or elow solid dshed ove elow < > Specil lines: Horizontl lines Verticl lines Shde in ove or elow Shde in left or right side > < x > x x < x shding ove ove elow elow right right left left line dshed solid dshed solid dshed solid dshed solid The solution to sstem of inequlities is the common re tht is shded. Exmple 6: Grph the solution to x Exmple 7: Grph the solution to x x LHS 0

12 Alger Summer Review Pcket VI. Fctoring Solved Exmples Fctoring (Common Monomil Fctor) Exmple 8: Fctor 5x z 0x z Solution: Find the gretest common fctor on the terms in the polnomil. GCF is 5 x z. Divide ech term in the polnomil the GCF to find the other fctor. 5x 5x z z 0x z x nd z 5x z 5x z 0x z 5x z x z Fctoring (Difference of Squres) Exmple 9: Fctor 6x 5 Solution: A term is squre if the exponents on ll vriles in it re even nd the coefficient is the squre of n integer. Both 6x nd 5 re squres. This will onl fctor if the two squres re sutrcted thus difference of squres. In prticulr, 6x = (x) nd 5 = (5). The inomil fctors s the sum nd the difference of the squres roots: 6x 5 = (x + 5)(x 5) Fctoring (Perfect Squre Trinomils) Exmple 0: Fctor x 0x + 5 Solution: A perfect squre trinomil is polnomil with three terms tht results from squring inomil. In other words, perfect squre trinomil = (some inomil). To fctor, we must first check tht the trinomil is perfect squre nswering the following three questions:. Is the first term squre? Answer: Yes, x = (x). Is the lst term squre? Answer: Yes, 5 = (5). Ignoring the sign t this time, is the middle term twice the product of x nd 5? Answer: Yes, 0x = (x5) Hving estlished tht the trinomil is perfect squre, it is es to find the inomil of which it is squre; its terms re in the prenthesis ove. x 0x + 5 = (x - 5) Since the middle term in the trinomil is negtive, the fctored result is sutrction. Fctoring (Product Sum) Exmple : Fctor Solution: If, s ove, the trinomil is in stndrd form, the product numer is the numericl term, 0, nd the sum numer is the coefficient of the liner term,. The unique pir of numer whose sum is nd product is 0 is nd 0. Therefore, the trinomil fctors s = ( + )( + 0) (Note: ll fctoring prolems cn e checked multipling out) LHS

13 Alger Summer Review Pcket VII. Simplifing Rdicls Solved Exmples HERE ARE THE FIRST TWENTY squre numers nd their roots: Squre numers Squre roots We write, for exmple, 5 5. "The squre root of 5 is 5." This mrk is clled the rdicl sign (fter the Ltin rdix = root). The numer under the rdicl sign is clled the rdicnd. In the exmple, 5 is the rdicnd. Properties of Squre Roots: Product Propert The squre root of product is equl to the product of ech squre root ; when nd Quotient Propert The squre root of quotient is equl to the quotient of ech squre root. ; when nd Simplif the following rdicl expressions. Exmple : 5 Exmple : Solution Solution 8 or 6 6 Rtionlizing the denomintor Exmple : Exmple 5: Solution Solution LHS

14 Alger Summer Review Pcket LHS Solved Exmples VIII. Simplifing Expressions with Exponents Integrl Exponents n times... n : se n: exponent ) ( 9 ) ( Rules of Exponents Product of Powers Quotient of Powers Power of Power Power of Product n m n m n m n m n m n m m m m Power of Quotient Zero Exponent Negtive Exponent m m m 0 n n n n Simplif ech expression. Exmple 6: 8 x x Exmple 7: 6 c c Solution Rule Powers Product of ppl 6x rerrnge the product 8 7 x x Solution Powers Rule Quotientof ppl 6c reduce frction c c Exmple 8: 5 ) )(0 ( Exmple 9: c c ) ( 8 Solution Solution 5 6 c c c Exmple 0: z x z x Solution = IX. Asolute Vlue Equtions nd Inequlities For this section, go to the link nd wtch the videos on solute vlue equtions nd inequlities.

15 Alger Summer Review Pcket I. Solving equtions in one vrile. Solve ech eqution.. (v ) 5 Exercises. 5n 5n. c c c c. 5c 7 II. Solving inequlities in one vrile. Solve ech inequlit.. 5 7x 7x x 5 x 5 0. x 8 x w 5. 0 w 5 LHS

16 Alger Summer Review Pcket Exercises III. Liner Equtions B. Finding the slope. C. Writing liner eqution. Find the slope of the line pssing through ech pir of points.. (0, 7) nd (, 9). (8, ) nd (-8, ). (-, -6) nd (-, ) Write n eqution of the line in A) slope-intercept form nd B) point-slope form with the given informtion.. slope =, -intercept = 7 5. slope =, (, ) 6. (, ), (-, -6) LHS 5

17 Alger Summer Review Pcket Exercises III. Liner Equtions D. Writing liner eqution of prllel nd perpendiculr lines. Write n eqution of line in slope-intercept form tht is prllel to the grph of ech eqution nd psses through the given point. 7. x ; (, ) 8. ; (, 6) Write n eqution of line in slope-intercept form tht is perpendiculr to the grph of ech eqution nd psses through the given point. 9. x 5 ; (, 7) 0. x 6 ; (, ) LHS 6

18 IV. Solving sstem of liner equtions. A. Grphing method. Alger Summer Review Pcket Exercises Grph ech sstem of equtions nd stte its solution..... LHS 7

19 IV. Solving sstem of liner equtions. B. Sustitution method. C. Elimintion method. Solve ech sstem of equtions using sustitution. Alger Summer Review Pcket Exercises 5. x x 7 6. x 8 x 7 Solve ech sstem of equtions using elimintion. 7. 5x 6 7 7x 5 8. x x 7 LHS 8

20 V. Grphing sstem of liner inequlities. Alger Summer Review Pcket Exercises. x. x x. x 6 x. x 5 5 LHS 9

21 Alger Summer Review Pcket Exercises VI. Fctoring x + 5x VII. Simplifing Rdicls VIII. Simplifing Expressions with Exponents x. 0.5x 0 5. x x z z 5 xz z 6. x 8x LHS 0

22 IX. Asolute Vlue Equtions nd Inequlities. Alger Summer Review Pcket LHS

23 Alger Summer Review Pcket I. Solving equtions in one vrile. p. Answers to Exercises. v = 6. c = 0. no solution. 5. = - 6. All Rel Numers II. Solving inequlities in one vrile. p... no solution.. 5. III. Liner Equtions p. B. Finding the slope. C. Writing liner eqution.... undefined slope. A) B) 5. A) B) 6. A) B) or III. Liner Equtions p.5 D. Writing liner eqution of prllel nd perpendiculr lines IV. Solving sstem of liner equtions. p.6 A. Grphing method.. solution is,. solution is,5 LHS

24 Alger Summer Review Pcket IV. Solving sstem of liner equtions. p.6 A. Grphing method.. solution is, Answers to Exercises. no solution IV. Solving sstem of liner equtions. p.7 B. Sustitution method. C. Elimintion method V. Grphing sstem of liner inequlities. p.8.. LHS

25 Alger Summer Review Pcket V. Grphing sstem of liner inequlities. p.8 Answers to Exercises.. VI. Fctoring. p.9. 5 ( 5 + ). ( + ). ( + 0)( 0). (x + )(x + ) 5. ( + 6)( + ) 6. (7 + ) VII. Simplifing Rdicls. p VIII. Simplifing Expressions with Exponents. p x 6 IX. Asolute Vlue Equtions nd Inequlities. p z x 5 6. z 5. -8, 8. No solution., All rel numers re solutions. LHS