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:AS:Jy@,:utc llt =O)r

- ocsvN (q iuc 691 =ocn6 (3A

: Ag : 3y (p iran lll = ortc

:bSv>)flt (c iurc qN8I =Jil= (CO'tilp (c :ruc glfZl : OE

- ocEVN @:wc lfVZ : Jy n

iwc Ar:)s (e'S'%gt=WQ7

:uc (g1,+y)e : Av'an t

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'*, *

({ :urc 97 lruc 91 luc 71

'%9t = oTod (qiytc ggl =f; lt

Modele detezesemestriale. ............111

Modele de teste pentru Evaluarea Nafionell ....................... 117

RECAPITULARE $I EVALUARE FINALA

Exercifiigiproblemerecapitulativepentrueveluareafinall.......... ........125ALGEBRA. ....".........,t2sG8OMETRIE................. ...............134

rNDrcATrr $r RASPUNSURr..........- ................139

=o+o5oLoO(

c)oUIoo

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o

'pIIqBIJB,{ nes ElnJsounJau clSetunu as rJEr '(raqll ue tttra} elSeutntt cs q lEt 'lalnJsott

-nJau trnluelJgooJ alse a) riuarcgaoJ cseurnu as g rS D elee.I eleJetunN 'BlncsounJau o nJ

l lnpur8 ap aliunca clSerunu es'0 + n rS g = q'D epurr'0 = 17 + fl2 tsIrIJoJ ep etience 6'S, r1o pzrclou as rrfnnce teuu rolttfnlos

eaurriln6 'Elelqlcpe clsc pinuriqo erirzodord 'plrp etienco Ltt JEurilu IecE nr eJnob^ou

-nJau purncoprl 'gcup arienia o rulued alfnlos clSeunu cs IBel Jputnu un

'reriunce 1e lderp lnJqLr.raur ap Eelrunuop pgeod le8e rnlnuuras etdearp uI slJcs else eo eecc

lur'rerfuncc ie Sugts Inlq,.rreru elSeurnu es le8e rnlnuures e3uets ur slJcs else eo BeeJ'(elncsounceu) eltqeue,n op Botlutnuap Eueod

' ' ''z ' t 'x:ufenrasqg

'9 + zE= (7 a z)g't!0=8-t7aIE'ZiZ+x- L-xZ'L

'(,,=") leia Jnuuas glup prn8uls o

aJude erur ug tollqupu,t ellnru ruru nus Bun nJ aJlluualuu rrirzodo-td luns agliencg

elnf,sounf,au o nf, I lnpp.r8 ap ulunf,l '[ @[arPrurl rrien:a ap

lln 'alep ttienlts roun e pltleuraleLU Pal

erEfult lliEnro op euetsts $ lDpnrlI lnlofldpc

Prqetlv

arerurl lrien:a ap auralsrsr$ rrien:a Joun earenlozar ul alualenrLlfa rolupturorsuell Ealeztllin

oretutl !lienra op aualsts nes ttfen:a tounro;rrinlos Earef,Uua^ nrlued alear araurnu nf, lnf,lel ap ro;t;n8at Eolpztl!ln '; .t

arerurl !lien)aap atlalsrs nes rienra urrd a;rqen;ozar alep rrienlrs taun ealelrytluapl

Se numeqte solufie a ecua{iei ar -t b= 0, unde tt, b € R qi a * 0, un numSr-r1, € 1R

pentru care propozitia axq | b = 0 este adev[rat[.A rezolva o ecua{ie inseamnd a determina toate solufiile sale. Acestc soiu(ii fornieazi

mullimea soluliilor ecua{iei date qi se noteazf, de reguld. cu S.

Dacd clupd o ecua{ie urmeazd o precizare de frrrmar e t,1, aoeasta indich nrullimea incare ia valori nccunoscuta. Se spune cd ecrn{ia dath cste deflniti pe mullimea ,('/ (sau cd sc

rezolvd in mullimea M. Dacitnu se face nicio precizare, se considera,4l- I{.

Num[rul 9 este soh.r{ic a ecualiei 2x - 7 -;r t 2 pentru ci, inlocuind in ecualie pe.r

cu 9, se obline o propozitrie adevdrat[; 2'9 -'7 - 9 + 2 (A) .

Orice numlr real este solu{ie pentru ecua{ia 3(z + 2) = 3z | 6: din aceastd cttuzd.

ecua{ia se mai numette qi identitate,Exista ecualii care nu au nicio solu{ie real[.

4(x - 3) = 4x + l0; 2z.r 5 = 2(: + 9) ctc.

Mulfimea solufiilor acestor ecualii este Z.

inlocuind necunoscutax cu numdrul 3 in ecua(ia3r ) 2 = 11, constatdm cI oblinem o

propozilie acleviratd: 3 . 3 + 2:11. Deci, num5rul 3 este solu{ie a ecua{iei. Putem spune

ie im rezolvat ecualia? Nu inc6, deoarece nu suntem siqtrri ci am at'lat toate solu{iile. S[presupunem cd numdrul a estc soiu{ie (gi el) a ecuafiei 3r + 2 : I1. Atunci. inlocuind

necunoscuta -t cu numdrul a, oblinem propozifia ndevdratf, (egalitatca) 3a t- 2 = 1 I ' Vom

scideadinarnbiimembri ai einumIrul 2,de uuderezulti ca3a+2 2=11 -2.adictr3a=9.Vom imp[rfi ambii rnembri cu 3 ;i oblinem u = 9 '.3. Deci, rz = 3.

Numai acum putem atln.na ca am rezolvat ccualia: ea are o singurh soltr{ie, 9i anumc

num6ru1 3. $i ecualial = 3 are ca soluqic doar nutr-rirul -1.

Deci, ecua{iile: 3r + 2 = 11 $i -r = 3 au aceca;i solulie. elt'trind echil'alente.Dou[ ecua(ii sunt echivalente in cazul in care au aceleaqi solulii O ecuafie simpla de fotma

r = d, unde a este numdr real dat, are ca solulie doar nttm6rul a. Arunci c6nd rezolvdm o

ecuaJie oarecare, incerclm sa gdsim o alta, de fornra -r = o, care s[ t-te echivalentd cu cea

dat5. Putem folosi unndtoarele reguli, care conduc la ecna{ii echivalente:1) Se pot trece tennenii diutr-un membru in celalnlt, scliimbAndu-le semnul.

2) Se pot inmul{i (imp[4i) ambii membri ai ecuaJiei cu nr.urere diferite cle zcro.

in general, o ecua{ie de forma ax+ b = 0, unde a qi D sunt numere reale (iar a*0),va fi numiti ecaalie de gradul I cu o necunoscutd.

O asemenea ecuaJie se rezolvd in doud etape:

1. Scddem din ambii membri pe b 9i ob{inem ax = -b.h

2. impdrtim ambii membri ct a gi obtinem x = -L. Aceastd ultimi ecualie are evident

hca unic6 soluJie numdnrl real - ''- 9i este echivalenti cu ecua{ia qx + b : A

6a

oI

HH

oov)oU:o.9+oEq)#g

=

oDacd a = 0 $i b = 0. anl

orice numdr real este solulie a ec

Dacd, a = 0, b + 0. atuncreste imposibil, deoarece prclclus,.

de zero. in general, ecua(iilc- nula aceasta folosind regulile care ,

Spunem cd dou[ numere reidacd se reprezintd in acelagi punr

Exemple:

l.Dacd,a=3 $i b=J9 ,atr:r

2.Dacd o=(2-16)' ti a=

Proprieti[ile rela(iei de ega

1. Reflexivitatea: x = x, pent

2. Simetria: dacdx = y, afi)nc

3. Tranzitivitatea: dacd x : r

Egalitatea se pdstreazd dacf,acelagi termen sau dacl inmul{irloc urmltoarele echivalente, nungi operatiile cu numere reale:

a=bee*x=b+x,(V)a-beq-x=b-x,(y)a-bea.x=b.x,(y)aa-bea:x=b:x,(V)a

Pe scurt, putem spune c[:o dacd se adun6, se scad :

egalitiji, se obJine tot o egalitate.

, - fa=boaca I . ahx

lc=d'Exemplu: Demonstrali cd daciAdundnd in ambii membri ai r

- 2*y = 2ry - 2xy, care este echirnumdr real este zeto doar cdnd nu

membri ai egalitdlii numdrul r.. rez

1.1. Ecnrvllnl1t

1.3. DE EGATITA]

1.2. Ecunllt DE GRADUT I cu o NECUNoscurA.

REDUCTIBITE I-A ECUATII DE GRADUT I CU O A

L'0=g+.D"{Enr

luepl^e eJe erlsnce pluFln ENBerv

'(0+D JBI) aluoJ eJaurnu luns g

'oJez ep ellJeJlp eJetunu n:'lnulues el-npllgqlulqJ{

:e1uelu,rrqce rriteec nc EluelsArgce erJ ps aJEJ '12:

o ruEAIozsJ prrgc rcuqY 'l2 Fu-su€uuoJ ep Eldulls erl€nce o'nlruos

r

'olualB^rqf,e pqg alr

erunu€ rS 'ellnlos pm8uts o arP r'E:o)

'6=rtEpcrpe'7- II = Z-Z--Dtlruo^'I I = z + Dg (eewFJP/sel qpurncolul 'rcunly 'll = Z + r€ !gg 'eprinlos eleol IBUB ruu gc r-mi

eunds ure1n4 'rerfence e erfnlos ao ureu{qo gc Iup}Bisuoc 'Il : Z -.

't

'gzfte) Els€ecB ulP !g + zE = (z -'(v)z-

r ed eriunce uI putncopl 'ec rutu:

'u :,r{ preprsuoJ a{

es pc nes)79 eerurfpru ed prruqefu1 esu{pru pc1pq elseece'14

= 's n3

pzeetilJoJ 11in1os elseov 'e1es ag!

UI =orJprunuun'g*prSr =

=o+o5o=oQ(

ao(Aoo

HHIo

',i - r ellnzat'tr Iruerunu lJiqtlle8e ru rrqr.ueuIIqIus ul prrgunpv '0 =,{- rEunzeJ'o;ez else lEp lluEtunu puEc JBop oloz elso IueJ ]Elunutnun 1n1e4qd eccrBoecl '0 = -({ - r) eoleirleSo nc qluelp^rllce else atec ,ix7

- ,{x7 = ,tx7 -- ,{ + ,x ealeyluBa rueuriqo 'txz- Iear lnrlurnu Iliqtlle8e r? r]qtuelu llqurB uI pugunpv

',.{ - r rcunte -.ix7

= _i a zr ecEp pc rierlsuorueqD .))

\O+P:0+;)1-=-l q 0l $I

P'Q='' o)

/,-q .)-rl.l p=.))I rJunle' i EJEp

l-+q t+D) Q=n)

:ure re 'snds IeJl[V 'e]E]rle8e o 1o1 euriqo es .riplrlu8e

Pnop ruqueur n3 ruqurcru gedttu os nES rseilnurul es 'pucs es 'Errnps 3s pcpp

:pc eunds ualnd'1rncs e4

'(O +r) y = r'r7'D (A)'x i Q = x : 0 e q - n

:(0+r) U Dr'g'0 (A)'x. Q =x. D e q - D

i6 = x'q'o (A)'r - Q =x -- D e q - D

i6 = r'q'D (A)'x + Q = x -l D <:) Ll - t:)

:oleoJ elelunu nc oprfe;edo rS

e1ult1e8e ep eriule-r oJluJ elellllqtlecftuoo ep {gleudo-rc{ elnunu 'eiua1e.r.rr1ra alareolErrun colnE gJIpV Inllsu lolJIlJ nn-.tlttt:d elcIle8c o (urrllEdu.{) nrrilnurur !,acp nu-( uerlrJel rselecelilttleia leun IE llqtu.rLtr rtqLrlrr rrrp (ruap!.s) urEunpr !rrp izi:.lsqcl as ealelrlefig

-Itr 3 z'd'r eouo nluad ? - r rJurrle, z - { rS { - x grlep :Belq! .rllzuurl .t

11 a d'raclrc n4uad ir-f rcmre ,{=xQczp:EIJFTTTIS .Z

lf, 3 rdruo nwad t= r :Bef8flalxoueu . I

:elper lolereunu eaur;[;nu ad ap1;p8a ap;a;[elar elllp]alldold'tt-t=

,(!N-z) ecereoep'q-onun1e, gtt-L=Q \s.(g^- z):o pceq.7

'€= !l ecer€oep'q-Dtau;nle' gf=g IS €= Dgceq.L

:a;duroxl'JoleJerunu exe ed lcund rSelece u1 gluzerder es pcep

ep8e lms q $ o el?er ererunu pnop pc rueundg

'cluelu^rqce lrience ul cnpuoc eruc elrln8er pursoloJ elsuec€ el

ecnpu ruolnd :L 1.!r: -q'.i'.ttls ErttroJ Qlseece qns elurzaJcl es nu aprfencc'lereuefl uI .oJez op

lueJrp ]e3r Jlurn'i ,-: .-^-i: :r.l \r,r.rz nJ leelJptunu rnunrcru lnsnpord sJeJUOep.lrqru6d-, a1r.eceeec'q --r0 !u:.::: - = .r.lllqclrel once{zodord rcun1e,6 tQ,0=Dgce1l

'rerlunce e arinyor- else Iuel Jpunu ecrJoIcep'0-r0euf,s as u = rr-:':.f:.irr o nrerlrzodo;drcunlu,g=17yS 0_Dlceq

;rrienlesqg

uoI

Yrn)soNnfrNonflrn

O C O octivitdti de ?nvdtore O O C

Determina{i valorile reale ale lui.r pentm care e ealitalile de mai jos sunt adevirate:

b)4x-1=9; c)2x-l:-9; d)61 5=7;D4x-3=-191. g)2x-9=-77; h)2x+13=5;j)3;r+7=16;' k) 1r- 1 =,r--+3; l)3x-2:.r-l-6in)-2x+5:11; o)5x+6:-14; P)-6x+11:)5.

b) 3(r 2) : t2 9i 3(-t + 5) - 33;

d) 3r + 24 = 6 Ei -2(-r l):22:l) 4x 13 : 1l Ei 7(-r + 3)- 63.

Stabili{i mul{imea soluliilor pentru t-recare ecuafie in parte:

a)3xf8=14; L',tl.r -l=r+l; c)3x+2=x-6; d)4x+3=x l5;e)3x 8=,r*-1: t)3r-11=x-23; E)4x+5='2x+73, h) 5x-!=1-x-t 1;

i)3x+ 1l--10: j) 7r+19 -__16; lQ5(x+3):-20; I)3x:r 18:

m)-6r*ll--10: n) llx-91 :30; o)4x+15: 5; p)Bx:xi 49.

Rezolr a1i ecualiile:

a)-9.r- 17= 10; b)3(x+2)=27; c)(n+2):3= 6; d)2(.t+ l)-3=5;e)7 2(x+3)=-ll; f)15+3(r 1)=6; g)](x-2) 13=8; h)6(1 3)+7= 35'

i)4 3(x+5):-11; j)(::r+ 1) : 5:5; k)3x-8- 13; l)-9+7r-5x+ I l;m)6x- 13:2x l: n)2,5x 3(1,5x+2):4,8; o)5x -9+2x:19;

I ^ I .3(r-5)p)2xt;:-0.(6): r)2x+1:-0.75---*; s) -- 5.r-18.'3232ArdtaJi cd urm[toarele ecua(ii sunt echivalente:

g) Determina{i numdrul realare solu{ia 3.

h) Determina{i numirul real: 7x + 3ax - 2 are solulia 2.7. Determinali valorile reale ale

a)aQ-x)-2=4+3(l-rb) 2(3x - 4) - 4a: x(a + t)c)3-x:2(x-5a)+ad)x-3(2x-s):s(.r-l)e)ax-l+2(-ax*1)=x-a

8. Rezolvatri ecuafiile in mullba) 4(x + 2) - 3x = -17;d) 32 -7(x - s): 3(_r - l);s) t4 - 8(x + 3): ?-x;

l6@-g+ 44:4(2t+ t)+bk) 3(x+ 2)-8:2(4x+31-1m)4(x+5)-5(x+3)=r-J;o) 3(2x + 3) - 2(3x + 2) = -31r)x-15+3(x+l)=5(r-l)

9. Determinafi valorile reale {rlente:

a)2(x-3)-3=x-2b)3(2x+1)-10=2(x-3)*c)2(2x-3)+ 19=3(x+4-d)2(x+3)-15=x*le)4x+17=3(x+1)+21f)3(x+ 5)- l1 =2(x+2)+3E)4x+2(x-5)=3(r+ t)-t

{ 0. Rezolvafi ecuaJiile in muti!a) 5(3x -2)-G(8x- 13)- l3rb) 2(3x - 4) + 5(6x - 7) _ 7r =c) 3(6x - 5) - 2(4x+ 13) + J -d)7x+4+2(x-5):t5+1-D2(x+3)-5(2x+ l): l2r-h) s(,r - 2) - 2x- (6 + 7r) :3(j) 6(r + 2) - 2(2x + 3) + 4r:3D5(r-2)+4x-3:6r+tt;n)5(x-1)+3r-4:3r+[;

{ {. Rezolvatri ecuaJiile in mu[ima)5x+9+3(2x-l)=Lr+21c)4x+7+5(x13)=2n+l:e)3r+9+2(3-x)=4+2(r+g) 5(2x- 3) + 3(5x - 6) + 13 =h)7(4x-l-aQx+8)+25=| 3@x - 5) - 3(2x + 7) + 32='.j) 6(r + 3) - 3(x + l) + 5 =-3r

a)2x+3=7;e) 3-r + 14 = 23,i)2r+5= 13;

nr)4x+7:31;

a)2r+l:7ti3-r 4:5;c) 7(x+ 1):6x 9i 3(.r+ 1) = ltt;e) 5("r + 4) : 25 9i -6(21 5) - 18:

oI

HH

oot/lI\))o.9"FoEq){-o€

a) Determina{i valoarea nurnarului real ll^ Stiind ca 3 este solu(ie a ecua{iei:

4(m+l)r 5ttx-1=2nt-6.b) DeterminaJi valoarea numdrului real m, gtiind ca -l este soluJie a ecua{iei:

3(m 2)* 2mx+9=5rrr-56c) Calculafi valoarea numdrului real m, pentru care 2 cste soiuJie a r-cuatiei:

1mx-3(2m + 5)r- 77 =4tn l,-.d) Aflati valoarea numirului re,al m, pentru care -3 este solulie a ecuafiei:

9mx + 8(3m 4)x + 18 nt = 35 - 6nt.

e) Determina{i valoarea numSrului real m, pentru care J este solufir- a ecuatiei:

3mx 2(3m 4)r + 13 i Jm = 14 + 8m.

a) Determinali valoarea reald a numdrului a. gtrind cd -3 este solu{ie a ecualiei:

4x'o(x +5)=la.Y+ l(.b) Determinali a e R, gtiind ci 4 este solu{ie a ecua{iei: a(7 x) 2x = 5ar i 26.

c) Determinali numdrul real a pentru care ecuaJia 2x + a: 4x ' 3 are solulia 2.

d) Determinali numarul real a pentru care ecuatia 2ax + 5(x 1):7r + 13 - 3ru are

solu(ia l.e) Determina{i numirul real a pentru care ecuatia a(x + 2) + 3(x 1) -- crx - 3 are

solufia 2.

f) DeterminaJi num5rul real a pentru care ecua{ia 2x a(r + 3) : 7 ar 'r 27 are solu{ia --5.

=a+o3o+.)'F(c)oIAoa

HHIo

:S + (t +rg)g =@Z- Dt-xV+ (e -rS)Z(liII + xE = @ - 9)Z- (E + rlr (p

:(t + x)Z+ 9 = (9 - x)Z -8 - rZ (q

rr-(Six)9+S-ra .- _

:(x f)I tl ,. .

's-'.\l rliy 116 ,- _ - J

:(g --r7)7 xt-=9 + (t +r)E -(e +r)9 (f:tt - (r * q)z: zt-r Q + xz)t - (E - ry)g (t:St - (e - r,)t = SZ + (S + rg)U - (g - xt)f Gte9 - t L- rE)8 = €I + (q - rS)e + G - xOS @

:G +x)Z r r:(:r i)Z+ 6-rE (o

:I +.rZ: (E + r)E t L + xV (cIVZ + xZ= (t - rZ)t -r 6 r_ r9 (e

:alear rolaJerunu ueur{1nu uJ elufence lierryoze1 .11:tI l-rg-, rt.i (t-r)EGr1111:r9-g xt+(Z r)9(l

:98 - xV+ (t + xdZ -(Z + r)S (l':(r x )t-@t-rq xZ-(Z-r)s(q

:6t xzt : i r xZ)S (g + r)Z G

'g- erinlos erc LZ + xD L : (t + +

or€ €-rn:(i-r)€+(Z+r)

ervDE-€I+xL:(j--x)9+r'7- e{n1os ere € + xb:D.

'92+r09-xZ-$- Do:1.

:rerfence e erfnlos else t-'w8+l

:rerience e erfnlos eNe ?-'rug + 91

:redence e erfnlos a'Lt -

:rerfence e e{n1os sls'9S

:rerfence u erfnlos elsa ;

'9-:rerfence e e{n1os a1s

'€9: (E a r)1 15 11 :5:ZZ:0 - x)Z- IS 9-=,i:eg : (S + r)g IS Zt:(Z

.8t_,r9:6foai6I:xZ+6-rS(o

iII +xS:xL+6-fi :gt:ig€-= L+(g-r)q (q i8 = €I -(Zig=E-(t+r)Z(p i9-=g:l

'6p -yx:xg(d :S-: S

i8I - r: n€ (l :OZ-:(ei1 +xE=6-rg({ itl+q=tigl-x=e+ry(p ig-x-

:euE'92-:It +r9-(d :Vt-:

i9 ax=Z-xtfi !t +x-ig= EI +xZ(q ;Ll-=i1= 9 - x9 (p i6-=

:etreJplepu luns sof reur ap a1

OOr

:V +(1-x)L= t+G-rt[-(t+rrSlZ(a !I +18 :G -x)Z +V +xL(p:gZ - @ - r)9 = 9 + (€I + xilZ -(S - rg)E (c

:Ot - (St + x)91 = xL - (t - xg)S + G - xdZ @,

?g-Ge-8)9r =xEr- (Er -rs)q -Q-xOs@qEaJ JolaJerunu uaurflruu u1 elufence fenyozeA .g,

gl -!=(I +r)f-(Z attt)x tS g + (I +x)g= (S -x)7 axp (E

TI +-T=1t.-ut[- rutS lS E +(Z+ x)Z= It -(S +r)g G. - - rrr)yrS

- -. . irrr; rs

tZ+(t+r)[=/t+rt(eI+r(:9i (e+DZ(P

:- :-LutZ t$ ot-(t+x)t=6t-r(s-rz)Z(o__ _- _ : r'rrzlrStt+(S-x)Z:gt-(I+rZ)E(q

z x:f (t-r)z(E:01Lra

-:' " -: -- - ,, rnlruqunu "3[8 JIUoJ ap]olel rleunuralecl

-- -:= :tt+(Z-r)g=(t+_r)g.r-9I-.r(r:- - : * :a+(t+x)t-=(.(,+rilZ-(E+rZ)e(o

lg -r= (e + r)S _. (9 1 r)7 (ur:8a-r!-,:_- : . . .-: :x€ (g+rt)Z_:g (Z+r)tt)j

r;y-'- : - - - irZ+Q+tZ)t-Wr(V r.)9(l'-, _ - ir7:(g1r)g tt(; irt:SI-r_-::.- , (9 i)r(e :(i r)g=(S x)/-Zt(p '.1+xZ:(i-r): . . (S+n)9(q :tt -jrt (Z+x)f e

: : :- ,oieJarunu eeurfltur u1 epLience lie.tlozaX'(r-9)i-: :- - :-.t[ rS D r=(t+rr)7, 1-rr(a

1')/., rs (t-r)s:(s-_tZ)s r(p. I l\o+(,tg-x)Z--r [(J

!I +crc(I-)r:(1 +r)g-r IS (t +rc)x:Dl-(V-xdZ(qiI :(l 1a)r- a rS (r- t)g * I = Z - (r- g)h (e

:eluelelrqce Ims eJBoryrrrun egtn€ apc nnued ., rnl ele el€eJ elrJolzl rieumuelaq .1.7e{n1os emZ_mt+xL:

: (t + xdj - De)7, - (Z + rXt - z)g q&r€ arec nqued rz Iuer lru?umu {euruueleq (q'g e{n1os a.e

(Z +,t)t - Z + xg : (t - fr)t +rg- s{Ence a.ruc n-Buad, IEer IruErunu fequuslag (t

r)_ir"; I) 5(r- 1)-15=35-4(x+2);Ir 5t--1r - 4) 7(4xt 9)+ 41-28 3(5r+ 6).

Rezolvali in mul(imea numerelor naturale ecualiile:a)4(x+3)-3(2x -r3)=7(r t 4)- l5(-r+ l) t 8;

b) 5(2x 3) 3(2x + l) 48 = 4(.r + 3) 414x r 7) + 30;

c) 5(3x+ 14)-4(5,u+ 1l) - l5 = 2(-r+ 38) - 4(2x + 13)- 11;

d) 9(ax 3) 7(3.x + 5) = 5(3.r - 10) 4(2x t 15) - 4;

e) 2x(x + 1) - 3(x r 4) = 2,t(,r 3) * 8.

Rezolvali in rnul{irnea rrurnerelor reale ecualiile:

a)7(x r2)-2(,u 4)=$(y+ )) 9; b) 10(r 1) - 3(4.r 7) - 2 = 4(3x 5) 'r 1 ;

c)3(2x 7)-5(x 2) 3=2.(r 7\+7; d)4(-t 5)+3(3r'-5):2(2x+5) 9:e)5(2x + 1)- 8(1 x)=2(3x I 2) t 5; f) -l(5 +:1t)-6(.r- t 2'\:5(2x t)-a;g) l8(2-r) t-6(7r 12)=5(2y - l) -3; h)2(3.r+4) 5(2.r -3)= 7(-t-3)+ 1l;1) 4(9x 4) .3(3.i -' l0) + I I ,= 4(2x - 21') - 5;j)2(5x-4) -6(4.i -3)+ 1l:6(3,1 1) -7

Rezolva{i in mul}irnea numerelor leale ecua[iile:a) 3(4x - 7) 4(5 - 3x) : 10r I 3;c) 3(2x 7) 5(x - 2):2x 4;

b)5(2-r 1) (15+19.r)-3,r 6(x+2);d),s(2x+ l) 2(3.r;+2)+3.r- l.l -5r;

oI

l-{l{

E'ovtoGlo.9F(,Eq)+(,=

10

e) -5(2x l) 3(t -x) 6(7x - t2).- 1s(8 r) - 84;

0 5(3 - 2-r) - 3x 2 6(2x - 3):2(3 4t) - 2(3r r 4);g) 4(9x - 4) - 2(1 -- 2x) - 7x : 5(x --22) 3( I 0 3"r) 4;h)2(sx- a) -(3"t+2) 3(5r- l0) :3r' -6(4x-3)- I l.

Rezolva{i ecr.ra(iile urmdtoare in mul{imea numerelor intregi:a)2[6(x+3)-3(r t5)] 4x=6(.r*lj; b)tx- 3[7(r-r3) - 5(.r r4)l:5(r r 1)lc) 2[5 + 3(r-' l)] =4(r + 7); d) 3l4,r 3(x t 5)l :2(2x 23) - 19

e)4[5(x 2)-3(,t 4)] r-s-x- l1 =6(-x tl) 1;

f) 2[ 8 - 4(3x 9)] - s = 3[19 - -5(2-r 3)] 21;g) 2U5 - 4(3x 8)l -'z = 3[1'1 - 5(2r - 3)] + 18.

Rezolvafi ecuafirie urmdtoare in mul{irlea numerelor reale:a)217(2:t l) - 3(4-t 3)l - 12 - s[l 2(2 -3x)]- t9;b) 19 - 3[2(2 + 5"r) 1l(?-r r 1)] .- 2[8 - 3(3 - 4.r)] r 6;c) 3[3(2:1 5) - 2(4x l)] + 26 - 41,{(-t + -j) 3("r + 2)l + 13;d)2[s(3r-a) a(2.r..9) 8] 7.-4 213(2x-5)+181:e)2{1x 3[3(,r+ 1) 2("r r 4)] 9] : 5(.r + 3);

0 (-, s){9-t -a[s(.r'--2)-3(,r 4)]+3] =x(.r+ 2) + 1.

Rezolva{i ecuafiile umitoare in mulfimea uumerelot' i'eale.

1t .-F1) .tl .1.r-4 1 a \ r-l 2x+l2l- I 14) I 5 ")'':l :

.3x+2 ^[q.-4 .i . 7.r-{.) rr-bc) --- - 2l

- -..... .,ii -) ! ) l) ) 1

,, 3.r+14 [: 1Zr-S -r+3)]_ 2(r+l) x-12.rll

----'-.1

i:..) 2 ,L4 r._3 _ 9 ))_ 3

_ 2_,

,. x*l 2x-t- 3(2r-S'' z - 3 =?[ l '

"l zlz*_r(3x+t_2r+r)lL \+ 3,')al :{r *11, + z( 2* -5x

+r''t 3L (' : )

, I t(2x+3 3x-2\l Ie' Lx-l[ 4 --;)1 ,

n z*-!(6*-t -sx-2)-:' 3( 4 3 )-lQ. Rezolvali in mullimea numen

4 Ji(*-z)+2Ji(x+l=afia1 zJ s (x + z) + aJ 6= :.,6(a

d 2*68 +Jr)-JrA*2x)=.ay :,(6+ Js )-,(:Vf -zf)q zJi(,+Je)-"(J: +r)= zJ

i s.ti(. - J a)- zG - r) =,(le) Js k + Jr o )- z.ti$ -.Jro

exersare **

II. z t's + sll I =l0rl * - e\z pz -(91' +

fgt ta :r - '=1r--[-q-=trl-"lr-(ur -\gr, o

: x - if L=(r + 9,A)r

-- (ll' * \91"2 r"

: (:r-e *,)9lc =(gl"z - !14" -($ * y;"9 1,

,(gr *,)El = (x7 + t)!l -(gl * !f),2 1"

: r!,lr-(gl z*rr)LN= 2ft+(z+x)977 1q

: gee =(r+r)9|z+(z-4!t (,:ufunce aleEqFrt aleu JoleJelunu eeurlfpur ug lie,rlozeg '97

i(gr\rz1z(tv\s 'L(l-;.'l-?l- * = lr- *- t- x, ) ,- *,

'*

,( , * t '.i_ z =gl( s _ o )!_."'l ," (r-gt S-{ I S+r tL\Z-"e t+xZ)t _] '

s fzl( e ) le I'53*' =t;-L['."--xz )t*,)Z* rii Q

./ t _ z ). tf t n \ I'(r+rz r*,)(t+")=[[,*17 - r;4)t-*z)z

@

.z_( 6_€)t_€_z,oS-rt [g+x S-xZ)E L-xZ T+x "c l(t z\ E1 v

=frL=Lt r-",**)' r-a)z-or+xr (B

ryD[rroru eeu{pru uI ereo}puun eyrfznce rfunlozea "gt,

c lv zl t t( z \.ll€+Iz=to-, lr.-r* r [e *", -xt))Jz te

i71+x [E (t )l e =t - xz+l1t* z I r * - xz

))t @

,l.r-( , * s )r-, . ,l!=l( s .- , )r--l-l!(" L - \r-er - e-q)e (or+rs)e _Jr [(z-xs t+*z)r (rr+r€X ]z

."

:--q-={-r-! [!- , (-!-.-.)li, t xz-s l*z Zl* "(e+* )))"':a1Ear.E[eJaunu eeurfynur uJ eJuolpuun eprfence lie,tlozeg .g j

z I z(z rt.l ' r-xb=r'l*-a [;.,."J.]

g

, s g l(t ) _l_ z : r*1,*rryr =

s1-,, * [[1*.r

-"2 )t+z )z +

r* * (e

=a+o3a=oQ(

ooI,,tao

HHIa

tet - (tZ- rdZ = [(; - rh

:(t + r)S =l(V + r)S - (S - tl.:6a-qu

.I I

:f:l

ixs-EI:xE+(Z+xilZ-lt:(z + x)g - xt: ("6t + St) - (

:tt +(g -x)L=(t-rdS-l:V-fi -xZ)S=(Z+r)q-(:e-G+tZ)Z=(9-r5)g-

:t +(E -xilV=Z-(L-ry)g-(

:t t - (ig

2{. RezolraF in mulqimea numerelor reale urmdtoarele ecua}ii:

a's x(t + zJi) + zJi (* + rJr) = Ji (zJ; -,) - r,g 2x(a+rJr)-zJi (zJ|-,) = 6(s, +:..6) -2e - 4x);

c1 x(t + +.6) + z(z + Ji) = zJilzx + r) - 3 ;

f--0 2xJ(J5*6)' -(rJi-J5), =*J2o +Jrx ;

e'1 +(zx+:15 ) - 2r(.'6 - l) = 2,6(, + -s) ;

D2x :Ji -+(.,6-.'),r--, L , l i'-..l

gr rvi(:-J,) r ri(: -Jr)' =.r/(r*Jr) -ri(:- Js)'

hr i(s *z"rr,)- :,r[l - 7u [ = r"5 -.,&x,l .f-

i r Ju ( -f, + r ) * r/1 "5 - rT )' : z,.r/1 r *,,Io )' - .,6 (.,8.. + r ),

- -l

r- r--_ '

jr zxut(r-J:) +ri(r-J:) =Jr(V, s) -,r/(J,-2)',

22. Rezolvali urmdtoarele ecua{ii in mulJirnea numerelor reale:

.4_i'+5 +(.tr-ll) l(-1-rr Q \'--r. li-9 l.r+13 .r+t{ 2=-14 2t b t+ iu 12 30 20 3

Rezolva{i in rnul{imea IR. ecua{iile:

3.r+7 1x-9 l2x +7 I 2r-.1 J.r+.'l 2-3-r' 5a)___ =__-l-__. b)_ =-- --lr0 l8 45 l0 -l -1 I 12

3x+14 5"r+i2 ;r+-38 3 2n+13 1l^\ -- =-------r45104520'

,r+3 x 2x+5 ;r+1 5^\ a -Ld,

-

246412'2.r+l r 3 ^ll r-J

-T

-

.I 6 I l: l

lr-5 r--i v-i461

- 2x-l -1,r'-3 - |f) i " ' =r.r*l-4-.. s)21 6'

lr+-J 3.r+l -lr+8 2L\

-r-=-----.

;\

r.5 20 ll 60

2x-l x+l I .r+l 7b)

-+---==---

+ -:20 l-q 4 l1 ll,.gr_7 2.t+.q 6vrll 5.u,' 1 6 14 2l

q:j 5r-e_l_7-10,.38t28'

-lr-2 l.r+5 I lx+5(t3248

r 2-r 1

--6=433, l(r+2) .r 5(x_-3) .^, *

-a_.=----r-.

oI

HH

ooI/)o():ci.(]+oEo)+g€

t?

24- Rezolvali in mulJimea Q en

.3x-5 x*4 5x-7 :

-='72t4:.2(3x+1) r*5 3x-2

15 3 5

,,6x-5 3x+1 4x-3*!

-

1263o x x+4 x-l x+3,t -a-5 l0 6 s

,, 2x-5 x-7 4x+11lt I

-

434

i) 3.r+ 4_ 5*- 9

- 3x+lo

-2425. Rezolva{i in mul}imea Q ecr

. 3x-5 5x+l 2x-6_ l=_2481r +'l

c) 4(x -2) + -:: -' =5(r*_il2

,4x-5 3x+2 x-l 1

-,

-

=--l-632:3(x+4\ 2x+9 I.r.

-

= i_tr?b' 7 3 -7\-

n., ,2_4(,r+l) _7x+4 _A'535., 3(2x+ l) 2(;r+5) 3r-j

7 6 tl26. Rezolvati in mullimea Q ecu

.lOx+3 l-3x 8x+{=__353

. -r' 3x-7 3-2x /a+.nl -I-f_=_'9 t8 t0 {5

.4x-9 3x-40 2x+5=_-'396,2x-3 2-3x 4x+3

ftl ___

3241., 3x+ 5 x+7 5--r

=t--'463

(p

(q

=o+o3o+o' o(c)o(Aoo

HHI

!

!t 0I 9 II I L+xZ 8-xE V-x9

0€99t0I '-r I f

-- L' b-xt Ll+xz L-xs

0l I Z E 01l-

I t+xz [+r l+r' \\

f.9t-_

-

il

r-9 ' L*.\' j- \[ZINZI.9 E+xV xE-Z €-xZ

8t 9 6 t .-tr -- \-E'S+)'a 0r_-rf 6-^'v

99V01816 . -!-=-

r- r- Ia

I 9+rZ xz-t L-xt .r

st9t s_+- +- (uI t+x8 .\[-t [+-[0t

:eprience fo eeunipu ug {e.t1ozag

9t 'z L-:-a-

"' (€-x)s t ' (t

tt '-r-I X_Z

SVZgg+xz I ' g+xz

. 8 _Zt_ 8 _XOI_ L I 6_15

,ZI Z

I re-a

E0z0t;tz 8+x EI+xz 6-r;

,rz_ vr _ 9 _I II+r9 6+r;ZIZI'SI '-f

- L I+x I Zr-r:3[eal

' .(t': -l.z'

: (1+ r71 \_

VIIZ(r + r)E

9L

, t -' S. = S *-:- .-lztu9+xz (t+x)z v+xL (l+r)i z- ". .t .L t L

:(€+r)_t_(E+r)TI= 6+*Z_ (r+rX

(ts

OI .Z 9 Z 9 Z Z T 9 .

-!lt'

_rl-. =---

lt q-TJ-=--- 15 ' t " 'l' G-*dV Z+xt L+xV " l' l-x Z+xt g-x7'ZZZZ :L-(S+x)s =7j;*(z+r)s (p :-- (g+x)s:

'*n*Q-x)v @

. o[ 91 9-'- 9 ro . z - 8 -r* I - ' ,n1+r-- I+r€=1t*r;t-a*ra tr f -t-- g+tz- t-I+rg s-x€ \"

:e[6snce fo eeur{1nul ul4en1oze5 .g7

Z ____v+xt0l+I[ 6-rs

ttv- ll+-\'t L-x 9-xz

0t s 9 0t Iltf

- 1+

!l [+.\' I-n l+x x'ttezl

-

\P[+r'9 [-rt l+)[ s-r'9

i st s f sl I

-

,1

: (l+1')[ Z-rS S+r (l+xt)ZZ VI Z L

.T- \Ux /_-r'l l+x E-.Yt

:ayrfence fo eaurflnur u1 rie,tlozeg

(f

: (xt-