ມ.2_ຄະນິດສາດ_Math

• d;f7no.9zqoG 10800 − x = 2(9300 − x) cmo7jk0v’ x = 7800 .llj q,zqo 9twfhG 10800 − 7800 = 2(9300 − 7800) 3000 = 2 × 1500 3000 = 3000 ສະຫງສ.ວວນ.ລິສຂະສິດ ltoAo ]k7k0v’xN, 7toyflkf c,oj 7800 du[D 8q;1jk’ @D 76z6h|b’j 1kpg0qhk\\q,.shoadIPoshv’|bj’ z6h]t $ dhvog0qhk|q,gs^nv16j10dhvoF 4hk1kp .shz6h]t5 dhvo g0qhk|q,9t[+r=15 dhvoD 4k,;kj oadIPoshv’oAo ,uma’\\qf9ad7qoL [qfcdhG • g]nvd8q;]a[ { ;k’.sh y c,jo9eo;ooadIPoma’\\qf0v’shv’oahog-yj’ y gxao9eo;o4h;o[;dD { 9eo;og0qhk|q,ma’\\qfmuj76oe,k1kpc,oj G ˜ 4y + 10 d=]tou1kpzh6|bj’ $ dhvo gs^nv !W dhvo ˜ 5y − 15 d=]tou1kpzh6|b’j & dhvo [+r= !& dhvoD • lhk’lq,zqo dko1kpg0Qk|q,ma’lv’d=]touF 9eo;og0qhk\\q,9t[+xjPocx’D gIqkwfhlq,zqo 4y + 10 = 5y − 15 • cdhlq,zqo 4 y + 10 = 5y − 15 4 y − 5y = −15 − 10 − y = − 25 y = 25 • d;f7no.9zqo0v’lq,zqoD 9kdG 4y + 10 = 5y − 15 cmo7jk0v’ y = 25 .llj q,zqo 9twfhG 4× 25 + 10 = 5× 25 − 15 75 + 35 = 125 − 15 110 = 110 4nd8hv’ ltoaho 9eo;ooadIPoma’\\qf0v’shv’oAoc,jo @& 7qoD   95

8q;1kj ’ #D 9q’j -vdsklk,9eo;o4h;o8yf8+daomuj,uzqo[;dgmqjkda[ ()D [qfcdhG • g]nvd8q;]a[ gvqo9eo;omu|bj’c,oj x • 8k,g’njvow00v’[qfg]dgIqk9twfh x + 1 c,oj 9eo;omulv’D • (x + 1) + 1 c,jo9eo;omulk,D vy’8k,g’nvj ow00v’[qfg]dG 9eo;omu|b’j T 9eo;omulv’ T 9eo;omulk, = 78 ltoAo gIqkwfhG x + (x + 1) + [(x + 1) + 1] = 78 x + x + 1 + x + 1 + 1 = 78 3x + 3 = 78 3x = 78 − 3 3x = 75 x = 25 9eo;omu|b’j c,oj 25 ສະຫງສ.ວວນ.ລິສຂະສິດ 9eo;omulv’c,oj 25 + 1 = 26 9eo;omulk,c,jo (25 + 1) + 1 = 27 zqo[;d0v’lk,9eo;oc,jo 25 + 26 + 27 = 78 gsao;jk4nd8hv’D ltoaho lk,9eo;o4h;o8yf8+daomuj,uzqo[;dgmqkj () c,joG 25 ; 26 ; 27 8q;1kj ’ $D l;o8vo|bj’gxaoI6[lucj 9lkdF ];’d;hk’gmqjk 3 0v’];’pk;D 9q’j -vdsk];’pk; 4 c]t ];’d;hk’0v’l;ofa’j djk;D I6h;jk ];’Iv[0v’l;ogmqkj 280m. [qfcdhG djvovnjo 8hv’I6h;kj G ];’Iv[0v’I6[lucj 9lkdgmqjklv’gmnjv0v’];’pk; [;d];’d;hk’D • ;k’.sh a c,jo];’pk; c]t 3 a ];’d;hk’0v’l;oD 4 vy’8k,g’nvj ow00v’[qfg]dG @×X];’pk; + ];’d;hk’? = 280   96

ltoAo gIqkwfhG 2 × ⎛ a + 3 a ⎞ = 280 ⎜⎝ 4 ⎟⎠ 7a = 560 a = 560 = 80 7 fa’j oAo ];’pk;c,oj 80 m c]t ];’d;hk’c,oj 3 × 80 = 60 m. 4 #D [qfg/ydsafD !D xuoU mhk; dymv’ ,uvkp5 !@ xuD r+0v’]k;,uvkp5 $! xuF 4k,;jk vud9adxu mhk; dymv’ 9t,uvkp5gmqjkda[vkp50v’r+]k;.oxt95[aoL @D 4hk76 c[j’xaog0Qk|q,xa’.shoadIPoc8]j t7qogmqjkMdaoD g,njvc[’j .shc8j]t7qo # czoj 9tgs^nv !@ czjoF 4hkc[j’.shc8]j t7qo $ czoj 9t[+r+ # czjoD 4k,;jk oadIPo,uma’\\qf9ad7qoL #D ohv’-kpwfhpkj ’vvd9kdgInvowxsk3I’IPo g-y’j ,uw]ptmk’ 2 km. !W okmu ສະຫງສ.ວວນ.ລິສຂະສິດ 8+,kvhkp wfh0uj]qf4u[8k,ohv’-kpwxD 9kdoAo 9adokmu vhkp9twxmaoohv’ c]t mao16j[vj o.fL g,njv Ih6;jkohv’pkj ’fh;p7;k,w; 80m / okmu c]t vhkp0uj]qf4u[ fh;p7;k,w; 240m / okmuD $D 9q’j -vdsk9eo;o4h;o[;dluj9eo;omugj xao9eo;o768j yf8+dao c]t ,uzqo[;d];,gmqjk *WL &D 7v[7q;0v’ mhk; lvow- ]hP’oqddk’cd c]t 38dt8jkp];,daoma’\\qf #W 38F 4hk oa[9eo;o8uo];,daowfhma’\\qf,u )$ 8uoF 4k,;jk 7v[7q;0v’ mhk; lvow- ]hP’oqddk’cd9ad38 c]t dt8jkp9ad38L *D ok’ l5lk7vo ,ug’yo !WDWWWdu[ ]k;-Ng0qhk9uj # dhvo c]t oq,c]ad8k-vp|bj’dvj ’ ]k 7k #DWWWdu[F g9qhk0v’Ihkomvog’yo.sh]k; &D&WWdu[D 4k,;kj ]k7kg0qhk9udj hvo|bj’9addu[L (D oadIPog7upIh6c]h; zqo[;d0v’lk,,5,.o0v’I6[lk,c9m5dMI6[ gmqjkda[ 180° ( x + 70)D  • 9qj’0Polq,zqogrjnv-vdsk,5,ma’lk,0v’ I6[lk,c9muj.sh,k0hk’]5,j oUD x D ( x + 20 )D   97

ສະຫງສ.ວວນ.ລິສຂະສິດ• 9q’j -vdskc8j]t,5,0v’I6[lk,c9fa’j dkj ;D )D ok’ 7erao ,uvkp5s^kpd;j kmhk; 7erPo # xuD zqo[;dvkp50v’lv’7qogmqkj da[ !( xuD dD 4hkmhk; 7erPo,uvkp5 x xuF ok’ 7erao 9t,uvkp59adxuL 0D 9q’j 0Polq,zqo grjnv-vdskvkp5 0v’ok’ 7erao c]t mhk; 7erPoD 7D ok’ 7erao ,uvkp59adxuL mhk; 7erPo ,uvkp59adxuL _D r+gInvozh6|bj’oe.-h|b’j lj;olk,0v’g’yogfnvo0v’8qogv’ grnjv9jkp7jkg-qkj gInvo c]t 7kj rkluS lv’l;j olk,0v’g’yogfnvomujpa’gs^nv9jkp7kj vksko c]t g7yj’|bj’0v’ g’yogfnvopa’gs^nv9jkp7jkg7jnv’o5j’ c]t dkoradrjvoD pa’gs^nvvud !WDWWW du[ ]k;mhvow;hD 9q’j -vdsk g’yogfnvo0v’r+gInvozh6oAoD !WD ,ulk,9eo;o c8]j t9eo;o]5f]njodao ! sq;|;j pD zqo[;d]tsj;k’lk,9eo;o fa’j dkj ;gmqkj da[ ###D 9qj’-vdsklk,9eo;ofaj’djk;D !!D zqo[;d0v’lk,9eo;ogmqjk 188D 9eo;omu|bj’s^kpd;kj 9eo;omulj v’ 12 c]t 9eo;omujlk,gmqkj lv’gmnvj 0v’9eo;omulj v’D 9q’j -vdsklk,9eo;ofaj’dkj ;D !@D 9q’j -vdsk9eo;o|bj’g,njvG dD 76o,aoda[ # c]h;]5f]q’ #W 0D sko,ao.sh # c]h;]5f]q’ #W 7D |bj’l;j olk,0v’,aogruj,.llj k,l;j oluj 0v’9eo;ooAo c]h; [;dda[lulj ;j oshk 0v’9eo;ooAo wfhlv’gmqjk0v’9eo;ofaj’djk;]q[vvd &*D !#D ];’Iv[0v’I6[lucj 9lkd\\b’j cmdwfh 44cm D 9qj’-vdskgoNvmu0j v’,aoS I6h;kj ];’pk;]noj ];’d;hk’ 3cm. !$D zqo[;d0v'lv’9eogmqkj 19S 9eo;omulv’s^kpd;jklk,gmnvj 0v’9eo;omu|bj’ 5. 9qj’-vdsklv’9eo;ofa’j dkj ;D         98

[qfmu @# vtltg|uzqoD !D dyf9tde 5 0,1 dt8jkp-vddyozad[Q’grnjv9t wxIvf[vj omu,j uzad[Q’ dyf9tdemu ! dt8kj p9t8hv’phkp]q’wx sk9eo;o muj|hvpd;jk 0 s^n 4hkphkp0Bowx8hv’phkpwx sk9eo;omusj ^kpd;j kD ‐1 ‐2 0 0,01 9qj’-j;pdt8jkp-vdglAo mk’wxdyozad[Q’D ‐1,5 ‐1,8 ‐0,1 ‐1,8 ‐1,1 3 ‐11 ‐2 ‐10 ‐1,8 ສະຫງສ.ວວນ.ລິສຂະສິດ ‐4 ‐6 ‐1,81  ‐1,79 ‐5 ‐1,8 ‐1,82 33 4 ‐2 5g 3 dyf9tde @G gruoj gvqk\\kd38ho-y’.ljzk’-y’ma’lv’ g[Nv’G dD 9q’j .sh0=hla’gdfg[y’j zk’-y’ma’lv’ g[Nv’c]h;0Po9eo;o,;olko.l[j jvo9Eg,afD .......+ ......> ......+.......   99

• 4hkgvqk\\kd38ho-y’  2g 8jn,.ljzk’-y’ g[Nv’]tvao 9tla’gdfgsaoco;.fL 9kdoAo 9qj’0Po.l[j vj o,u9ED 5 + 3 + ......... .......3 + 3 +........... • 4hkgvqk\\kd38ho-y’ 3g vvd9kdzk’-y’g[Nv’]tvaoD la’gdfgsaoco;.fL 9kdoAo 9q’j 0Po.lj[jvo9Eg,afD 5 + 3 − ......... .......3 + 3 − ........... 0D zkj o7e8v[0hk’gmy’oAo oadIPo9qj’.sh0=hlts^5[D dyf9tde #G gryjo.sh vtltg\\uzqo a < b S 9qj’0Po.l[j vj o9Eg,affh;pdko76oma’lv’rkd0v’vtlt g\\uzqoda[9eo;ofP;daomuj.sh.o;q’,qo 4hk;jk 7kj 0v’ a c]t b gmqkj da[G  dD a=4  0D a = −6  7D a = −6 b=6 b=4 b = −4 ສະຫງສ.ວວນ.ລິສຂະສິດ   0 × 4  …  0 × 6 …  … …  0 …  … …  1 − 1   2  2 2×4 < 2×6 …  >  …   2  ‐2 …  …  …  c > 0 a<b c < 0   …  … …    100

@D .97;k,D !D oypk,G vtltg\\uzqoc,oj lv’leo;o s^n lv’9eo;omu[j +gmqkj dao 8yf8+daofh;pg7njv’\\kp < ; > ; ≤ s^n ≥ 8q;1jk’G 5 > −8 S −7 < −5 S a+b<c • vtltg\\uzqomuj,ug7jnv’\\kp7ndaogvUo;jkG vtltg\\uzqomyffP;daoD 8q;1kj ’G • a > b c]t c > d • a < b c]t c < d • a ≥ b c]t c ≥ d • a ≤ b c]t c ≤ d @D 75o]adltot0v’vtltg\\uzqoD ສະຫງສ.ວວນ.ລິສຂະສິດ .sh a; b c]t c c,jo9eo;o9y’ gIqkwfhG dD 4hk a < b c]t b < c gIqkwfhG a < c 8q;1kj ’G 2 < 4 c]t 4 < 5 gIqkwfhG 2 < 5 0D 4hk a < b gIqkwfhG a + c < b + c 8q;1kj ’G 2 < 4 gIqkwfhG 2 + 3 < 4 + 3 7D 4hk a < b gIqkwfhG a − c < b − c 8q;1jk’G 2 < 4 gIqkwfhG 2 − 3 < 4 − 3 ’D 4hk a < b c]t c > 0 gIqkwfhG ac < bc 8q;1kj ’G 2 < 4 gIqkwfhG 2×3 < 4×3 { 4hk a < b c]t c < 0 gIqkwfhG ac > bc 8q;1jk’G 2 < 4 gIqkwfhG 2×(−3) > 4×(−3) 9D 4hk a<b c]t c>0 gIqkwfhG a < b c c   101

8q;1jk’G 2<4 gIqkwfhG 2 < 4 3 3 { 4hk a<b c]t c<0 gIqkwfhG a > b c c 8q;1kj ’G 2<4 gIqkwfhG 2 > 4 −3 −3 #D dkoxP[mP[9eo;omqfltoypq,lerafD dD dkoxP[mP[9eo;omqfltoypq, muj,u7kj gxao9eo;o[;d7ndaoD { 4hk9eo;o.fskd,urkdlj;o4h;os^kpd;jk 9eo;ooAod+s^kpd;kj D 8q;1kj ’G 4,56 > 3,989 grkt;jkG   4,56 ⎫ 4 > 3 3, 989⎬⎭ { 4hklv’9eo;oskd,urkdlj;o4h;ogmqjkdao gIqkg[yj’38g]dmqfltoypq,0v’lv’ 9eo;oF 4hkskd7ndaogIqkg[y’j 38g]dmulj v’ c]t 8+MwxDDD 4hk9eo;o.fs^kpd;kj ສະຫງສ.ວວນ.ລິສຂະສິດ 9eo;ooAod=s^kpd;jkD 8q;1jk’G 7, 254 < 7,312 grkt;kj G 7, 254⎫ 3 > 2 7, 312⎬⎭ 5,145 < 5,16 grkt;kj G 5,145⎫ 4 < 6 ⎬ 5,16 ⎭ 0D dkoxP[mP[9eo;omqfltoypq,mu,j u7kj gxao9eo;o]q[7ndaoD { 4hk9eo;o.f,u7jkle[6os^kpd;kj 9eo;ooAo9t|hvpd;jkD 8q;1kj ’G −5, 41 < −4,89 grkt;kj G −5, 41 = 5, 41⎪⎫ g-y’j 5, 41 > 4, 41 −4, 89 = 4,89⎬⎪⎭ 7D dkoxP[mP[9eo;omqfltoypq,leraf Xmu,j ug7njv’\\kp7ndao s^n 8jk’dao? fh;pdkola’gdfg7njv’\\kp0v’zqo]q[D { 4hkzqo]q[]ts;j k’lv’9eo;o skd,ug7njv’\\kp[;d s^n gmqkj da[l6o c,jo 9eo;omugj xao8q;8A’]q[9ts^kpd;jk s^n gmqjkda[9eo;omugj xao8q;]q[D   102

9eo;omqfltoypq,leraf a c]t b F 4hk a − b ≥ 0 c,jo a ≥ b D { 4hkzqo]q[]ts;j k’lv’9eo;o skd,ug7njv’\\kp]q[ s^n gmqjkl6o c,oj 9eo;o mujgxao8q;8A’]q[9t|hvpd;kj s^n gmqjkda[9eo;omujgxao8q;]q[D 9eo;omqfltoypq, a c]t b F 4hk a − b ≤ 0 c,jo a ≤ b D 8q;1kj ’G !D 10, 25 > −15 grkt;jkG 10, 25 − (−15) = 25, 25 > 0 @D −0,09 < −0, 08 grkt;jk −0, 09 − (−0, 08) = −0, 01 < 0 #D [qfg/ydsaf !D 9q’j xP[mP[9eo;o]5j,oUG dD 0,408 c]t 0,507 S 5,3 c]t 5,302 338,39 c]t 33,839 S 0,004 c]t 0,0039 0D − 21,0034 c]t − 21,0099 S 0,99 c]t 0,101 338,39 −1,001 c]t −1,00099 S −1,019987 c]t 1,0199799 @D 9q’j xP[mP[c8j]t76j,;olko]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ dD 3210 g c]t 3, 210 kg 0D 2k g 7 g c]t 2,07 g #D 9qj’9af]Po9eo;o]5j,oUc8j|hvpsks^kpD dD 6 ; 5 ; 14 ; 7 ; 0 ; 1 7 7 7 7 7 7 0D 4 ; − 1 ; 15 ; 22 ; -8 ; 11 11 11 11 11 11 11 $D 9qj’-vdsk9eo;omuj|hvpd;kj !D dD 3 ; 11 ; 5 ; 2 ; 14 ; 1 ; 10 ; − 15 ; − 17 5 9 8 8 9 8 9 8 9 { 9kdoAo 9q’j 9af]Po9eo;ogmy’oU 8k,]efa[c8j|hvpsks^kpD &D 9q’j 0v[0aho9eo;o - 7,436 fh;plv’9eo;omqfltoypq,mu1j j64afda[,aoD *D 9q’j c8h,cdo9eo;omuj[ao95 A,B,C, D,E c]t F 1j6]5j,oU7noD 9kdoaho9qj’0Po9e o;o8+wxoU.lj[jvo,u9E.shg\\ktlq,G 0,7 ; 1,5 ; 0,5 ; 0,6 ; 0,25 ; 1,6   A BCD 1 EF 2 0  … … …  … ……   103

(D gryjo.shvtltg\\uzqo 5 > 2 4hkgryoj [;dma’lv’rkd0v’vtltg\\uzqoda[ ( s^n {( vtlt g\\uzqo.\\mj uwj fhIa[9tgxaoco;.fL )D 4hkgryoj 76oma’lv’rkd0v’vtltg\\uzqo −2 < 4 da[9eo;o 3; − 3; 0. vtltg\\uzqo.\\mj uwj fhIa[9tgxaoco;.fL _D 4hkgryoj 76oma’lv’rkd0v’vtltg\\uzqo − 1 < 1 da[  − 2 vtltg\\uzqo.\\jmujwfh 2 2 3 Ia[9tgxaoco;.fL !WD gryjo.sh a ≥ b 4hk;kj a + c ≥ b + c 9qj’-vdsk7jk0v’ c D !!D gryjo.sh a ≤ b 4hk;jk ac ≤ bc; ac ≥ bc c]t ac = bc 9qj’-vdsk7kj 0v’ c .oc8j]td=]tougmy’D !@D 9q’j 76oma’lv’rkd0v’vtltg\\uzqoda[9eo;o16j.o;q’g]a[ c]h;.sh0=hla’gdfg[yj’ vtltg\\uzqoxjPocx’co;.fL dD −7 < 5; (5) 0D −4 < 6; ⎛1⎞ 7D 8 < 10; (2,1) ⎝⎜ 2 ⎟⎠ !#D 9q’j 76oma’lv’rkd0v’vtltg\\uzqo da[9eo;omuj16j.o;q’g]a[c]h;.sh0=hla’gdf ;kj ສະຫງສ.ວວນ.ລິສຂະສິດ vtltg\\uzqoxjPocx’co;.fL dD 5 > 2; (−3) 0D −6 > −8 ; (−8) 7D −4 < 4; (−0, 2)       104

[qfmu @$ vtlq,zqo0Ao|bj’mu,j u|b’j 8q;]a[ !D dyf9tde dyf9tdemu ! .ocdo9eo;o9y’F 9eo;omuj16gj [Nv’-hkp 9t|hvpd;kj 9eo;omuj16jg[Nv’0;kD ‐10  ‐5  O  5  10  gIqkwfhG −10 < −5 < 0 < 5 < 10 { 3fpvy’.lcj do9eo;o9y’F 9qj’8nj,g7njv’\\kp < <, > s^n = < .l[j vj o9Eg,afD a. −18.... − 20 b. 245....(−9)(−5) c. 3....5 d. 0,323334....0,343434 f . − 35.... −10 − 25 e. (3 − 24)....(4 − 36) g. 18....16 h. ⎡⎣5 − (−6)⎤⎦.... − 9 ສະຫງສ.ວວນ.ລິສຂະສິດ j. 4x....3x, g,nvj x < 0 i. 4x....3x, g,nvj x > 0 dyf9tdemu @ 9qj’la’gdfdkocdhvtlq,zqo]5j,oU rhv,ma’ltcf’.9zqo.lcj do9eo;oD dD x + 3 > 8 x + 3 + (−3) > 8 + (−3)                           x > 5 .9zqo0v’vtltg\\uzqo c,oj m5dM9eo;o9y’ x musj ^kpd;kj 5 D { .9zqo0v’vtlq,zqo lk,kfltcf’.lcj do9eo;ofa’j oUG     ‐2 ‐1 0 1 2 3 4 5  6 7 x 0D 2x − 5 ≤ 3 2x − 5 + 5 ≤ 3 + 5 1 × 2x ≤ 1 × 8 2 2 x≤4 ‐2 ‐1  0 1  2 3  4 5 6 7 x 10  5

7D 9qj’cdhvtlq,zqo]5,j oU rhv,ma’ltcf’.9zqo.ljcdo9eo;oD • x + 5 < 10 S 2x +1≥ 3 • − 2x +3 < x + 6S 3− 2x ≥ x !D .97;k, !D oypk,D vtltg\\uzqo ax + b > c; ax + b < c; ax + b ≥ c s^n ax + b ≤ c g-y’j xtdv[ fh;p9eo;o|bj’mu[j +maoIh6 gvUo;jkG vtlq,zqo0aho|b’j mu,j u|bj’8q;]a[D a; b; c gxao9eo;omuIj h6c]h; c]t a 8jk’l6o (a ≠ 0) x c,jo9eo;omuj[+maoIh6 g-yj’gvUo;jkG <8q;]a[<D 7kj 0v’ x mu8j v[lt|v’da[ vtlq,zqogvUo;jkG <.9zqo<D @D dkocdhvtlq,zqo ສະຫງສ.ວວນ.ລິສຂະສິດ gryoj oe.-h75o]adltot0v’vtltg\\uzqogrjnvcdhvtlq,zqoD .od=]touvtlq,zqogxaog]d lj;o c]t r6fgxao9eo;o 8hv’0Bor6fI;j ,djvo c]h;xtr6fD 8q;1jk’G !D 9q’j cdhvtlq,zqo x + 3 > 8 [qfcdhG x + 3 > 8 [;dma’lv’rkd0v’vtlq,zqoda[ (−3) G x + 3 + (−3) > 8 + (−3) gIqkwfhG x > 5 .9zqo X7kj 0v’ x ? lk,kfltcf’ .lcj doj 9eo;ofa’j oUG    ‐2  ‐1  0 1 2 3 4 5 6  7 8 9 10 ............... x  s^n lk,kfltcf’fh;pd5,j .9zqo S = ]5; + ∞[ @D 9qj’cdhvtlq,zqo 1 x + 2 < 2 x+3 3 5 10  6

[qfcdhG 1 x + 2 < 2 x + 3 3 5 gIqk8hv’0Nor6fI;j ,djvoF .o[qfg]doUgIqkgvqkr6fI;j ,c,jo !& 9twfhG 5× x + 15× 2 < 3× 2x + 15× 3 5× 3 15 ×1 3×5 15 ×1 5x + 30 < 6x + 45 15 15 15 15 xtr6f gIqk9twfhG 5x + 30 < 6x + 45 [;dma’lv’rkd0v’vtlq,zqoda[ (−5x) + (−45) 9twfhG 5x + 30 + (−5x) + (−45) < 6x + 45 + (−5x) + (−45) −15 < x ltcf’;kj 9eo;omusj ^kpd;kj {!& c,oj .9zqo0v’vtlq,zqo0hk’gmy’     ............... ‐18 ‐17 ‐16  ‐15 ‐14 ‐13 ‐12 ...............ສະຫງສ.ວວນ.ລິສຂະສິດ x s^n d5,j .9zqo0v’vtlq,zqogmy’c,oj S = ]−15; + ∞[ #D 9qj’cdhvtlq,zqo −2x + 3 ≤ x + 6 [qfcdhG xt8y[af7n8q;1kj ! c]t @ gIqk9twfhG −2x + 3 ≤ x + 6 −3x ≤ 3 x ≥ −1 lk,kfltcf’.9zqo.lcj do9eo;o fa’j oU    ………….. ‐3 ‐2  ‐1 0 1 2 3 ………….. x s^n d5j,.9zqo0v’vtlq,zqogmy’c,jo S = [−1;+ ∞[ ly’j mu7j ;ogvqk.9.ljG { [k’vtlq,zqo[+,u.9zqogxao9eo;o9y’D { [k’vtlq,zqo,u.9zqoc,jom5dM9eo;o9y’D 10  7

8q;1kj ’ !D x + 3 < x − 5 g,njvcdhvtlq,zqo 9twfhG 0x < −8 \\kp7;k,;kj 0 < −8 c,jogxaowx[+wfhle]a[m5dM7jk0v’ x mujgxao9eo;o9y’D faj’oAo x + 3 < x − 5 [+,u.9zqo gryjo0Po S = φ 8q;1jk’ @D x + 5 > x + 2 g,nvj cdhvtlq,zqo 9twfhG 0x > −3 gIqkwfh 0 > −3 le]a[m5dM7kj 0v’ x mugj xao9eo;o9y’D fa’j oAo x + 5 > x + 2 ,u.9zqoc,jom5dM9eo;o9y’ gryoj 0Po S = \\ #D [qfg/ydsaf !D 9q’j cdhvtlq,zqo]5,j oU c]h;ltcf’.9zqo.ljcdo9eo;oD ສະຫງສ.ວວນ.ລິສຂະສິດ dD −3 < 3x 0D −6 < x − 7 7D x − 8 > 6 ’D x − 5 > −5 9D −3x + 7 < x + 2 lD −x > 9 -D −2x +11 < −4 pD −3( x +1) > 12 fD 3x − 6 > 27 8D 3x +10 < 25 4D 8x +15 > 73 mD −18 − 5z > 52 oD 14 − 4x ≥ 38 [D 3x −12 < 15 xD 5x − 8 ≥ 12 zD 14 − 2x ≤ 24 − 6x @D 7e4k,7nda[0=h ! dD 3(6 − 5x)≤ 12x − 36 0D 2(7 − 8x) ≥ 14x − 46 7D 9(m + 2) > 9(m − 3) 'D 6x −13 < 6( x − 2) 9D −6(2x −10) < 180 lD −7 (3x − 7) > 280 -D 2 − 3x ≥ 7(8 − 2x) +12 pD 3x − 2 ≥ x + 2( x + 5) fF 3⎡⎣4y − (2y − 7)⎤⎦ < 2(3y − 5) 8D 6 ⎣⎡5x − (3x −1)⎦⎤ ≥ 4(3x − 7) 4D −5(4z +1) < 23 mD 12 < 2(3n +1) + 22 10  8

oD 2( x − 3) + 7 < 21 [D 4( x − 2) − 6 > 18 xD 17 + 2( x +1) ≥ 12 zD 12 − 2(3x −1) ≥ 6 + 3(4 − 2x) #D 7e4k,7nda[0=h !D dD x −1 < − 9 0D x − 7 ≥ 2 10 8 9 7D x − 2 ≤ 4 ’D 11 ≥ 4 + x 3 6 5 9D 7x − 2 > 3 − x lD 3x < 3x − 4 − x 2 7 2 4 5 -D x − x + x < 2x −1− x − 1 + x +1 pD 3 ( x −12) > x − 24 2 3 2 3 5 fD x + 2 − x − 3− x ≤1 8D 2 − 3( x −1) + 5( x +1) ≥ 1 5 6 12 4 2 8 12 4D 2x − 3(3− x ) ≤ 0 mD x + 4 + x−4 ≥ 0 5 7 21 10 oD 3 − 2x +1≤ x+5 [D 2n − 5 − 2 ( n+ 1) ≥ n 8 8 2 3 4 $D 9q’j -vdsk7kj 0v’ x .oc8j]td=]tou]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ dD 3ax + 2b < 2ax − 8b, a < 0 0D 7ex − 2c > 5ex + 6c, e > 0 7D −5bx +12c ≤ 13c + 5bx, b < 0 ’D ax − 3b ≥ b − 3ax, a > 0 &D 9q’j -vdsk7jk0v’9eo;o[;d n grnjv.sh −5n +11 > −9 *D 9q’j -vdsk7jk0v’ x grnjv.shleo;o]5,j oU ,u7kj gxao9eo;o[;d dD x − 7 0D −x + 8 7D x + 2 ’D x − 5 9D −x + 3 lD 2x − 4 -D x + 3 − x pD 2x + 3 − 4−x 2 4 5 10 (D 9q’j -vdsk7jk0v’ x grnjv.shleo;o]5,j oU ,u7jkgxao9eo;o]q[D dD −x − 4 0D x + 7 7D x + 3 ’D −( x +1) 9D 2x +10 lD 4x − 2 -D x+5 pD − 2x +3 10 5 10  9

)D 9qj’-vdsk7kj 0v’ x grnvj .shleo;o]5j,oU gxao9eo;o[;dS gxao9eo;o]q[D dD 3x − 4 0D 2−x 3 7D −4 − x ’D 1 ( x − 5) + 1 (3 − x ) 3 3 4 _D 9qj’la’gdfI6[c8h,]5j,oU c]h;-vdsk7jk0v’ x grnjv.sh];’Iv[0v’I6[lujc9lkd |hvpd;kj ];’Iv[0v’I6[lk,c9ltg\\uD 4  x x ສະຫງສ.ວວນ.ລິສຂະສິດ 11  0

[qfmu @& dkocdh39fdPj ;da[vtlq,zqo0aho|b’j muj,u|b’j 8q;]a[ !D [qfg]d8q;1kj ’ 8q;1kj ’ !D .sh9eo;o|b’j F 4hkgryoj gvqk 2 0v’9eo;ooAo,k]q[da[ 1 0v’9eo;ofa’j dkj ; 3 4 9tgsao;kj zqo]q[0v’r;d,ao|hvpd;kj 120D 9q’j -vdsk9eo;ooAoD ;ymucdGh dD 7afg]nvd8q;]a[D ;k’.sh x c,oj 9eo;omu8j hv’dko-vdD 2 0v’9eo;ooAo9tc,jo 2 x 3 3 1 0v’9eo;ooAo9tc,oj 1 x 4 4 0D lhk’gxaovtlq,zqoD ສະຫງສ.ວວນ.ລິສຂະສິດ 8k,g’njvow00v’[qfg]dgIqk9twfhG 2 x − 1 x < 120 3 4 7D cdhvtlq,zqoD 2 x − 1 x < 120 3 4 4× 2x − 3× x < 12×120 8x − 3x < 1440 5x < 1440 x < 1440 5 x < 288 fa’j oAo 9eo;omuj8hv’dko-vdc,oj m5dM9eo;omu|j hvpd;kj 288 D   111

8q;1jk’ @D [=]ylaf0qolqj’lv’csj’wfh8yfxhkp[=]ydkofa’j oUD [=]ylaf d [=]ylfa 0 55000du[ + 630du[ / km 60000du[ + 610du[ /km 4hk8hv’dko1kd.-h[=]ydkoF 9qj’7yfw]jg[yj’;kj 7kj [=]ydko0v’[=]ylaf d 9txt1afສະຫງສ.ວວນ.ລິສຂະສິດ d;kj 7kj [=]ydko0v’[=]ylaf 0 9ad km L ;ymucdhG dD 7afg]nvd8q;]a[D ;k’ x c,oj w]ptmk’muj9twx 7yfw]gj xaody3]c,af(km)D [=]ydko0v’[=]ylaf d 55000 + (630× x) [=]ydko0v’[=]ylaf 0 60000 + (610× x) 0D lhk’gxaovtlq,zqo 8k,g’nvj ow00v’[qfg]dgIqk9twfhvtlq,zqo 55000 + 630x ≤ 60000 + 610x 7D cdhvtlq,zqoD 55000 + 630x ≤ 60000 + 610x 630x − 610x ≤ 60000 − 55000 20x ≤ 5000 x ≤ 250 fa’j oAo 7kj [=]ydko0v’[=]ylaf d xt1afd;kj 7kj [=]ydko0v’[=]ylaf 0 .ow]pt mk’wdl5fc,oj 250 km # \\kpgsfG [qfg]dgmy’1kdltcf’.shgIqkIh6;kj 4hkw]ptmk’|hvpd;jk 250km c,oj [=]ylad d xt1afd;jkD c8jw]ptmk’s^kpd;jk 250km 0Nowxc,jo 7kj [=]ydko0v’[=]ylaf 0 9txt1afd;jk 7jk[=]ydko0v’[=]ylaf d g-ajo;jk 4hkw]ptmk’ 251km gIqk9twfhfa’j oUG   112

  [=]ylaf d [=]ylaf 0 55000 + 630 × 2 51 60000 + 610 × 2 51 55000 + 1581 30 60000 + 1531 10 213130 213110 la’gdfgsao;jk [=]ylaf 0 xt1afd;kj [=]ylaf d c,jo 213130 − 213110 = 20 fa’j oAo 4hkgIqk9twx]5f 250km 7;og]nvd.-h[=]ydko0v’[=]ylaf d 4hkgIqk9t wxdkp 250km 7;o9tg]nvd.-h[=]ydko0v’[=]ylaf 0 @D s^addko];,.odkocdh[qfg]d39f !D vjko[qfg]d7nos^kpgmnjvgrnjv.shg0qhk.9lkpdPj ;rao]ts;jk’0=h,6omugj ryoj .shda[ສະຫງສ.ວວນ.ລິສຂະສິດ lyj’mujgryoj 8hv’dko.shgIqk-vdskD @D g]nvd8q;]a[8k,me,tfkgIqkg]nvd8q;]a[ c,oj vkdkomujgryjo8hv’dko.shgIqk -vdsk X g[y’j 7e4k,7no.shfu;kj gryjo8hv’dko.shgIqk-vdskspa’L ? #D lhk’vtlq,zqo8k,g’nvj ow0[qfg]d39f $D cdhvtlq,zqomulj hk’wfh &D lbdlk7no.9zqomuj-vdwfh .shg\\ktlq,c]h;8v[ #D [qfg/ydsaf !D Ihko7hkcsj’|bj’xtdkf]k7k0kp oq,c]ad8k-vp faj’oUG 4hk-Nc]ad8k-vp]k7k 25000 du[ 8+|b’j 3s^ 9t[+wfh0v’c4, 4hk-Nc]ad8k-vp]k7k 26500 du[8+|b’j 3s^ 9twfh0v’c4, ! xJv’ Ih6;jk ]k;0kppvj p.o]k7k 2500 du[ 8+|b’j xJv’D 4k,;kj 4hkgIqk8hv’dko-N oq,c]ad8k-vp 2 3s^ gIqk7;o9tg]nvd]k7k.fxt1afd;jkL @D 5 gmnjv0v’9eo;o|bj’g,njvgIqkgvqk,k]q[da[ 3 0v’9eo;ooAogsao;jkzqo]q[,u7jks^kp 2   113

ສະຫງສ.ວວນ.ລິສຂະສິດd;jkzqo[;d0v’9eo;ooAoda[ 20 D 9qj’-vdsk9eo;ofa’j dkj ;L #D g9qhk0v’Ihkov5xtdvodkolbdlkcs’j |bj’-NxN,0Po,k 500 sq;F g,njv0kpwxwfh9e o;o|b’j ]k;gsao;jk xN,0Pogs^nv[+g4y’ !WW sq;D 4k,;jk ]k;wfh0kpxN,0Powxc]h; 9adsq;L $D ok’ 85d8k-N\\kd]t,5f9eo;o|bj’D s^a’9kd]k;dyo\\kd]t,5fc]h; 5 |j;pF gsao;jk \\kd]t,5fmu]j k;-N,kgs^nv[+Ivf !W |j;pD 4k,;jk \\kd]t,5fmuoj k’ 85d8k-N,k ,uma’\\qf9ad|j;pL &D 4hk[;d #$ da[lv’gmnvj 0v’9eo;o|bj’gsao;jk|hvpd;kj lk,gmnjv9v’9eo;ooAoD 9qj’-vdsk9eo;ofa’j djk;D *D 9qj’-vdsklv’9eo;o4h;o8yf8+daoD I6h;jk zqo[;d0v’r;d,ao|hvpd;kj ##D (D 9qj’-vdsklk,9eo;o4h;o768j yf8+daomu,j uzqo[;ds^kpd;jk )*D )D 9qj’-vdsk7jk;afcmd0v’lv’,5,g-y’j zqo[;d0v’r;d,ao16]j ts;jk’ 85D sk 90D S 4hk;kj 7jk;afcmd0v’,5,mulj v’s^kpd;kj 7kj ;afcmd0v’,5,muj|bj’ 4D D _D .oI6[lk,c9F zqo[;d0v’lv’0hk’8hv’s^kpd;jk0hk’mulj k,D 4hkI6[lk,c9 ABC 0hk’ BC = 4 c]t AC = 8 − AB D 9q’j -vdsk 0hk’ AB D   114

[qfmu @* lq,zqomu,j u7kj le[6o !D dyf9tde dyf9tdemu ! .o0Ao,afmtpq,xumu ! r;dgIqkwfhIPodjP;da[7kj le[6o9eo;o|bj’D 7kj le[6o9eo;o.f|b’j c,jo9eo;ooAogv’mugj vqkg7njv’\\kpvvdg-aoj ;kj G −2,8 = 2,8 S 0 =0 S +3 =3 3fpvu’.lj7;k,Ih6gmy’oahoF 9q’j 7yfw]7j jkle[6o 0v’9eo;o8kj ’M ]5j,oUG • −12,05 S − 9 S + 2,25 • 5−3S 5+3S 5×3 • 6 − 1,5 × 3 S −8 × 2 S − 24 ÷ 2 dyf9tdemu @ gIqkIh6;jk lv’9eo;odq’dao0hk,dao,u7kj le[6ovaofP;daoD • 9q’j 0Po9eo;omujg\\ktlq,8n,j .l8j k8t]k’]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ x 1,8 0 2, 4 9eo;odq’dao0hk, x −7 3,5 9 x −5, 04 − 1 8 3 x 2, 7 14 0 9 dyf9tdemu # 3fpvu’.ldj yf9tde1 c]t dyf9tde 2 gIqk9twfh x = 2 c,jo x = − 2 s^n x = 2 grkt;kj − 2 = 2 c]t 2 = 2 9kdoAo 9qj’-vdsk7jk0v’ x mujgIaf.sh dD x = 3 0D x + 1 = 2 7D 3 − 2x = 5 ’D x + x = 2 11  5

@D .97;k, !D 7jkle[6o0v’9eo;o a = ⎧a; a≥0 ⎩⎨−a; a<0 8q;1kj ’G • 5 =5 • − 8 = − (−8) = 8 • x − 2 = x − 2 F 4hk x − 2 ≥ 0 • x − 2 = − ( x − 2 ) = 2 − x F 4hk x − 2 < 0 @D lq,zqomu,j u7jkle[6o ສະຫງສ.ວວນ.ລິສຂະສິດ dD lq,zqo.oI6[Ikj ’ x = a .9zqo0v’lq,zqo9tc,oj −a c]t a 8q;1kj ’G • x = 3 gIqk9twfh.9zqo0v’lq,zqoc,oj x = − 3 c]t x = 3 grkt;kj − 3 = 3 c]t 3 = 3 D • x = 0,75 gIqk9twfh.9zqo0v’lq,zqoc,jo x = − 0,75 c]t x = 0,75 grkt;jk − 0,75 = 0,75 c]t 0,75 = 0,75 • x = −1 lq,zqo[+,u.9zqogrkt;kj 7kj le[6o0v’9eo;o.f|b’j [+,u7jkgxao9e o;o]q[D • x = 0 lq,zqo,u.9zqoc,oj 0 grkt;kj 0 = 0 0D lq,zqo.oI6[Ikj ’ x + a = b Xb gxao9eo;os^kpd;j k 0 ? ;ymucdhG •x+a=b • x + a=−b ☺ 7;k,\\kpmk’fhkog]0k7toyf0v’lq,zqo x − a =b x c,jova[-yf0v’g,af|bj’mujsjk’9kdg,af A(a) fh;pw]ptgmqjk b D    x A(a) x bb 11  6

8q;1kj ’G 9q’j cdhlq,zqo !D x + 1 = 3 ;ymucdhG • x +1 = 3 • x +1 = −3 x = 3−1 x = − 3−1 x=2 x = −4   ltoAo .9zqo0v’lq,zqoc,oj x = 2 c]t x = − 4   −5 −4 −3 −2 −1i 0 1 2i 3 4 i  B A C la’gdfgsao;kj G B(−4) c]t C(2) c,oj lv’g,afmusj jk’9kd A(−1) fh;pw] pt|bj’gmqkj da[3sq;|j;p @D x −3 = 2 ;ymucdhG ສະຫງສ.ວວນ.ລິສຂະສິດ • x−3 = 2 • x−3 = −2 x = 2+3 x = −2+3   x=5 x =1   ltoAo .9zqo0v’lq,zqoc,oj x = 5 c]t x = 1    −2 −1 0 1i 2 3i 4 i5 6 7   BAC g,af B(1) c]t g,af C(5) c,oj lv’g,afmusj kj ’9kd A(3) fh;pw]pt|bj’ gmqkj da[ 2 sq;|;j pD #D 2x − 3 = 5 ;ymucdhG • 2x−3 = 5 • 2x−3 = −5 2x = 3+5 2x = −5+3 2x = 8 2x = − 2   x=4 x = −1 11  7

s^n gIqkvkf9txPj olq,zqo.shdjvo gxaoG x − 3 = 5 2 2 • x− 3 = 5 • x − 3 = − 5 2 2 2 2 x = 5 + 3 x = − 5 + 3 2 2 2 2 x = 8 x = − 2   2 2 x=4 x = −1 .9zqo0v’lq,zqoc,jo x = −1 c]t x = 4   −2 −1i 0 1 i 2 3 4i 5  B AC g,af B(−1) c]t g,af C(4) c,jolv’g,afmusj jk’9kd A ⎛ 3 ⎞ fh;pw]pt |b’j gmqjk ⎜⎝ 2 ⎟⎠ da[ 5 sq;|;j pD 2 ສະຫງສ.ວວນ.ລິສຂະສິດ #D [qfg/ydsaf !D 9q’j cdhlq,zqo]j5,oUG • x = 4S x = 3 S x = 3,5 S x − 3 =5 2 4 @D 9qj’cdhlq,zqo]5j,oUG • x −1 = 5 S 3x −1 = 8 S x = 3 S 2x + 7 = 3 S 3x −8 = 9 #D 9qj’cdhlq,zqo]j5,oUG • 5x −1 = 1S x −1,8 = 2 S x +11 = 6 S 7x + 2 = 4 S 6x −1 = 12 $D 9q’j 0Poxt3spd]5,j ouh gxaolq,zqomu,j u7jklq,zqo dD lv’g,af A c]t B ,uw]ptskj ’9kdg,af M (0) gmqkj 3 sq;|j;p 0D lv’g,af A c]t B ,uw]ptskj ’9kdg,af N(1) gmqjk 4 sq;|;j p 7D lv’g,af A c]t B ,uw]ptsjk’9kdg,af P(1) gmqjk 5 sq;|;j p 11  8

[qfmu @( vtlq,zqomu,j u7kj le[6o !D dyf9tde dyf9tdemu ! 8q;1jk’G gryjo.sh x < 2 \\kp7;k,;jk w]ptskj ’9kdg,af 0(0) G Ð wxmk’g[Nv’0;k[+g4y’lv’sq;|j;p Ðwxmk’g[Nv’-hkp[+g4y’lv’sq;|;j p g-y’j ltcf’fh;pG ‐4  ‐3  ‐2  ‐1  0  1  2  3  4  x ltoAo 7kj 0v’ x 9t16j.os;kj ’ −2< x < 2 D 3fpvu’.lj8q;1jk’0hk’gmy’F 9qj’-vdsk 7jk0v’ x muj8v[lt|v’D ສະຫງສ.ວວນ.ລິສຂະສິດ dD x < 1S 0D x < 2 dyf9tdemu @ 8q;1kj ’G gryoj .sh x > 4 \\kp7;k,;kj w]ptsjk’9kdg,af 0(0) G Ð wxmk’g[Nv’0;ks^kpd;jk $ sq;|;j p Ðwxmk’g[Nv’-hkps^kpd;jk $ sq;|j;p g-y’j ltcf’fh;pG ‐6  ‐5  ‐4  ‐3  ‐2  ‐1  0  1  2  3  4  5  6  xx ltoAo 7jk0v’ x c,oj x > 4 s^n x < − 4 D 3fpvu’.lj8q;1jk’gmy’F 9q’j -vdsk7kj 0v’ x mu8j v[lt|v’D dD x > 3 S 0D x > 5 11  9

dyf9tde # gryoj .sh x −1 < 3 \\kp7;k,;kj w]ptsjk’9kdg,af A (1) wxmk’g[nhv’-hkp c]t g[Nv’0;k[+g4y’lk,sq;|;j pD g-yj’ltcf’fh;pG ‐6  ‐5  ‐4  ‐3  ‐2  ‐1  0  1  2  3  4  5  6  A x ltoAo 7kj 0v’ x c,oj −2< x < 4 dyf9tdemu $ gryjo.sh x + 2 > 4 \\kp7;k,;jk w]ptsjk’9kdg,af B( − 2) wxmk’g[nhv’-hkp c]t g[Nv’0;ks^kpd;kj lusj q;|;j pD g-y’j ltcf’fh;pG ‐7  ‐6  ‐5  ‐4  ‐3  ‐2  ‐1  0  1  2  3  4  5  6 ສະຫງສ.ວວນ.ລິສຂະສິດ B x x ltoAo 7kj 0v’ x c,jo x < − 6 s^n x > 2 @D .97;k, !D vtlq,zqo.oI6[c[[G Ð x < a gIqk9twfh −a < x < a Ð x ≤ a gIqk9twfh −a ≤ x ≤ a Ð x > a gIqk9twfh x < − a s^n x > a Ð x ≥ a gIqk9twfh x ≤ − a s^n x ≥ a 8q;1kj 'G 9q’j cdhvtlq,zqomuj,u7jkle[6o]5,j oUG !D x < 5 gIqk9twfh −5< x <5 d5,j .9zqoc,jo S = ]−5;5[ 12  0

@D x ≤ 2 gIqk9twfh −2≤ x ≤ 2 d5j,.9zqoc,jo S = [−2;2] #D x > 5 gIqk9twfh x < −5 s^n x > 5 d5,j .9zqoc,jo S = ]−∞;−5[ ∪ ]5;+∞[ $D x ≥ 1 gIqk9twfh x ≥1 s^n x ≤ −1 d5,j .9zqoc,oj S = ]−∞;−1]∪[1;+∞[ @D vtlq,zqo.oI6[c[[G Ð x − a < b gIqk9twfh a −b < x < a + b Ð x − a > b gIqk9twfh x > a + b s^n x < a −b Ð x − a ≤ b gIqk9twfh a −b ≤ x ≤ a + b Ð x − a ≥ b gIqk9twfh x ≥ a + b s^n x ≤ a −b 8q;1jk’G !D x −7 < 3 gIqk9twfhG −3+ 7 < x < 3 + 7 1  2  3  4  5  6  7  8  9  10  11  12  13  4 < x <10 A ສະຫງສ.ວວນ.ລິສຂະສິດ x d5,j .9zqoc,jo S = ]4;10[ @D x + 2 ≤ 3 gIqk9twfhG ( − 2) −3≤ x ≤( − 2) +3 ‐7  ‐6  ‐5  ‐4  ‐3  ‐2  ‐1  0  1  2  − 5≤ x ≤1 A x d5,j .9zqoc,jo S = [−5;1] #D x +1 > 4 gIqk9twfhG x > ( −1) + 4   s^n x < ( −1) − 4 x > 3                       x < − 5    d5j,.9zqoc,oj S = ]−∞;−5[ ∪ ]3;+∞[ ‐7  ‐6  ‐5  ‐4  ‐3  ‐2  ‐1  0  1  2  3  4  5  6  xB x $D x −5 ≥ 1 gIqk9twfhG x ≥ 5+1 s^n x ≤ 5−1 ‐8  ‐6  ‐4  ‐2  0  2  4  6  8  10  x ≥ 6        x ≤ 4   x Bx d5j,.9zqoc,jo S = ]−∞;−5]∪[5;+∞[   12  1

#D [qfg/ydsaf !D 9q’j cdhvtlq,zqo]5j,oUG dD x <8; x < 0, 7; x < 7 2 0D x ≤5; x ≤ 5, 2; x ≤ 3 4 @D 7e4k,7n0=h! le]a[G dD x >18 ; x > 5, 4; x > 8 5 0D x ≥11; x ≥ 2,3; x ≥ 4 5 #D 9q’j cdhvtlq,zqo]5j,oUG dD x −1 <8; 0D x + 5 < 4 7D x − 5 >4; ’D x − 7 > 2 2 lD x −1 ≥5 9D x + 2 ≤3; ສະຫງສ.ວວນ.ລິສຂະສິດ -D x − 0,5 ≤ 3 ; pD x + 2 ≥ 0, 7 4 $D 9q’j -vdsk9eo;o4h;o x muj8v[lt|v’da[vtlq,zqofh;p;ymucmo7kj D dD 7 ≤ x ≤ 9 0D 1< x − 2 < 4   &D 9q’j cdhvtlq,zqo]5,j oUG dD 2 x < 5; 3 x < 7   2 0D 3 x > 2; 1 x >1  4   7D 0,8 x ≤ 1; 4 x ≤ 3  5 ’D 0, 4 x ≥ 3; 7 x ≥ 5 3 3 12  2

rkdmu $ I6[lucj 9F I6[lk,c9 c]t ;q’,qo [qfmu @) I6[lujc90hk’0t|ko !D dyf9tde A B .oI6[lujc9 ABCD F 0hk’ AB / / DDDDDDDDDDD O c]t DDDDDDDD / / BC c]t ,ul6odk’g7y’j 7n DDDDDDD grnvj 1kd-Uc9’;jkG AB = DDDDDDDDDDDDDDDDDD D C gryjoxt8y[affaj’oUG A c,jog,afg7y’j 7n0v’ DDDDDDDDDDDxP[.lj O c]t B g7y’j 7nda[DDDDDDDDDDDDmP[.lj O D gIqkwfh mvj o-nj AB g7yj’7nda[ DDDDDDDDDDDmP[.lj O \\kp7;k,;kj AB = DDDDDDDDDDDD 9q’j oe.-h;ymuxt8y[af0hk’gmy’grnvj -Uc9’;jkG dD AD = BC 0D O c,jog,afg7y’j dk’0v’ AC \\kp7;k,;jk OA = OC 7D O c,jog,afg7yj’dk’0v’ BD \\kp7;k,;jk OB = OD 9qj’oe.-h75o]adltot0v’,5,lts^a[.ogrnjv-Uc9’;kj G AnBC = nADC; nDAB = nDCB ສະຫງສ.ວວນ.ລິສຂະສິດ @D .97;k, B !D oypk,G I6[lujc90hk’0t|koc,oj I6[ A lujc9muj,u0hk’g-y’j \\hk0t|kodao8k,c8]j t76jD 8q;1kj ’G 4hk;kj ABCD c,oj I6[lucj 9 DC 0hk’0t|ko gIqkwfhG AB / /CD c]t AD / /BC @D 75o]adltotG dD lv’glAogoaj’9v, 0v’I6[lucj 90hk’0t|ko 8af dao16gj ,afg7yj’dk’0v’c8j]tglAoD A B 8q;1jk’G .oI6[lucj 90hk’0t|ko ABCD F lv’glAo O goaj’9v, AC c]t BD 8afdao16jg,af O gIqkwfhG OA = OC c]t OB = OD DC 123

A B 0D lv’0hk’g-yj’|hkdaogmqjkdao8k,c8]j t76jD O 8q;1jk’G 4hk;jk ABCD c,oj I6[lucj 90hk’ 0t|koF gIqkwfhG D C AB = DC c]t AD = BC A B 7D ,5,g-y’j |hkdaogmqkj dao8k,c8]j t76Dj D C 8q;1kj ’G 4hk;jk ABCD c,oj I6[lujc9 0hk’0t|koF gIqkwfhG Al = Cl c]t Bl = Dl #D l6odk’g7y’j 7nD B I6[lucj 90hk’0t|ko,u|b’j l6odk’ g7y’j 7ng-yj’c,oj g,af8afdao0v’lv’ A O glAogoaj’9v,0v’,aoD ສະຫງສ.ວວນ.ລິສຂະສິດ D C g,af O c,oj l6odk’g7y’j 7n0v’I6[ lujc90hk’0t|ko ABCD. $D dkolhk’I6[ 8q;1kj ’ !D 9q’j lhk’I6[lujc90hk’0t|ko ABCD 3fp;kj AB = 7 cm, AD = 4cm c]t AC = 10cm D 9kdoAo -Uc9’;jkG DnAC = BnCA D C 0=hlq,,5f AB = 7 cm, AD = 4cm F 4 cm 10 cm AC = 10 cm A 7 cm B 0=hlts^5[ lkh’I6[lucj 90hk’0t|ko ABCD ;ymulhk’G { lhk’mjvo-nj AB = 7cm { lhk’lv’mvj odq’j muj,ug,af.9dk’16j A c]t B ]aflt\\ugmqkj 10cm c]t 4cm F lv’mjvodqj’8afdao16jg,af C D { lhk’lv’mvj odqj’mu,j ug,af.9dk’16j A c]t C ]aflt\\ugmqkj 4cm c]t 7cm F lv’mvj odq’j 8afdao16gj ,af D D { 0uf8+ B sk C F C sk D c]t D sk A 9twfhI6[lucj 90hk’0t|ko ABCD muj8hv’lhk’D 124

8q;1kj ’ @D lhk’I6[lucj 90hk’0t|ko EFGH g-y’j lv’glAogoaj’9v,0v’,ao 8afdao16j O 3fp;jkG EO = 3cm, FH = 7 cm c]t EnOF = 110D xE 110D Fy 0=hlq,,5f EO = 3cm, FH = 7 cm c]t EnOF = 110D O 0=hlts^5[ lhk’I6[lucj 90hk’0t|ko EFGH H G ;ymulhk’G { lhk’,5, XnOY = 110D { ;k’g,af E c]t F 16gj lAo-nj [OX) c]t [OY) 8k,]efa[ 3fp;kj OE = 3cm c]t OF = 3,5cm { cdpj k;glAo-nj [EO) c]t [FO) { ;k’g,af G c]t H 16j [EO) c]t [FO) 3fp;jk OG = 3cm c]t OH = 3,5cm { 0uf8+ E sk F, F sk G c]t G sk E 9twfhI6[lucj 90hk’0t|ko EFGH muj8hv’lhk’D A ສະຫງສ.ວວນ.ລິສຂະສິດ #D [qfg/ydsaf !D .oI6[lk,c9 ABC [ao95I6[lucj 90hk’0t|ko9adI6[L D E X[+oa[I6[fvd9ao? H @D gryjo.shI6[lucj 90hk’0t|ko ABCD muj,ul6odk’ G g7y’j 7n O D9qj’0ufglAo-nj d1 c]t d2 0t|koglAo-nj J BD 3fpzkj o9v, A c]t C 8k,]efa[S CF I B 0ufglAo-nj d3 c]t d4 0t|koglAo-nj AC zjko9v, B c]t D 8k,]efa[D glAo-nj d38af d116jg,af I c]t 8af d2 16gj ,af J S glAo-nj d4 8afda[ d116jg,af L c]t 8afda[d2 16gj ,af K D 4k,;kj I6[muwj fh,k ,u9adI6[lucj 90hk’0t|koL 7nI6[.fcfLj #D .oI6[[lk,c9 ABC glAo9v,dk’ BE c]t CF 8afdao16jg,af G D { 9q’j ;k’ L c]t K 16lj j;o0uf8+0v’glAo-nj (GF) c]t (GE) 3fp;kj G FL = GF c]t EK = GE D { -Uc9’;kj BLKC c,joI6[lujc90hk’0t|koD $D dD |kplk,g,af A, B c]t D 3fp;jkG AB = 5cm, AD = 4cm c]t BnAD = 70o 125

0D oe.-hw,h[aomaf c]t ;q’;Po grnvj lhk’g,af C 3fp.shI6[ ABCD gxaoI6[ lucj 90hk’0t|koD 7D 3fp[+8hv’.-h[aomafcmd,5,F 9q’j deoqf7kj 0v’,5,8jk’M0v’I6[lucj 90hk’ 0t|ko ABCD D &D lhk’I6[lujc90hk’0t|ko EFGH g-y’j I c,jog,af.9dk’.oc8j]td=]tou]5,j oUG dD EF = 8cm c]t FG = 5cm 0D EF = 8cm, EH = 5cm c]t Hl = 60D C F 7D EF = 8cm, EH = 4 cm c]t EG = 7 cm E ’D HG = 6cm c]t EH = 4cm *D .shI6[lv’I6[lucj 90hk’0t|ko B O ABCO c]t DOFE Xfaj’I6[?D D 9q’j -Uc9’;jkG Bl = El A (D -vdsk7kj 0v’ x c]t y .oc8]j td=]tou]5,j oUG dD 2x + 3 0D 25 28 7 2y -8 O 3x+ 4 9 3y -5 ສະຫງສ.ວວນ.ລິສຂະສິດ 7D 80D 35D 36 5xD(6 y + 5)D ’D 4x -y 24 2x +16 )D I6[lucj 90hk’0t|ko ABCD, M c]t N c,jog,afg7yj’dk’0v’0hk’ AB c]t DC 8k,]efa[D 9qj’-Uc9’;kj G I6[lujc9 AMND c,oj I6[lucj 90hk’0t|koD _D I6[lujc90hk’0t|ko ROSE, W, X, R XO W Y c]t Z c,oj g,afg7y’j dk’0v’ IR, I IO, IS c]t IE 8k,]efa[D 9q’j -Uc9’ EZ YS ;jk I6[lucj 9 WXYZ c,joI6[lucj 90hk’0t|koD 126

[qfmu @_ I6[lujc9lkd !D dyf9tde dyf9tdemu ! 9qj’les^;fg[yj’;kj .oshv’IPo,ulyj’0v’vao.fmujgxaoI6[lucj 9lkdL dyf9tdemu @ la’gdfg[y’j I6[ c]h; 8v[7e4k,D A B dD c8]j t,5,0v’I6[lucj 9 ABCD c,oj ,5,spa’L O 0D 9qj’[vd[aofk760j v’0hk’mu0j t|kodaoDDDDDDDDDDDD { 0hk’muj,u7;k,pk;gmqjkdao DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D C 7D I6[lucj 9 ABCD ,u9adglAogoa’j 9v,L { 9q’j .-h;q’;Po c8h,;q’,qomu,j ug,af.9dk’16gj ,af8afdao0v’lv’glAogoa’j 9v, c]t ]aflt\\uc,oj OA. oadIPola’gdfgsaoco;.fL { 9qj’xP[mP[7;k,pk;0v’ OA, OB, OC c]t OD. { 9qj’-Uc9’;jkG AC = BD ສະຫງສ.ວວນ.ລິສຂະສິດ dyf9tdemu # AI B 8afg9hp.shgxaoI6[lujc9lkd ABCD, I, J, K c]t L gxaog,afg7y’j dk’ KO L 0v’0hk’ AB, DC, AD c]t BC DJ C 8k,]efa[ Xfa’j I6[? 9kdoAo ra[8k, Ivp(IJ) c]t (KL) D { fh;pdkora[8k,Ivp (IJ) , gIqkwfh g,af A g8a’g,af DDDDDDDDF g,afDDDDDDDDD g8a’ g,af D c]t 0hk’DDDDDDDDg8a’ BCD glAo-nj (IJ) c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 0v’lv’0hk’ BC c]t AD. { fh;pdkora[8k,Ivp (KL) , gIqkwfh g,af D g8a’da[DDDDDDDDF g,afDDDDDDDD g8a’da[ B c]t 0hk’ DC g8a’da[0hk’DDDDDDDD glAo-nj DDDDDDDD c,jocdjo g7yj’7n0v’0hk’ DC c]t DDDDD fa’j oAoF I6[lujc9lkd ABCD ,uDDDDDDDDDDDDDDDcdog7y’j 7nD 127

@D .97;k, B !D oypk,G I6[lucj 9lkdc,jo I6[lujc90hk’0t|kog-yj’m5dM A ,5,0v’,ao]h;oc8jgxao,5,lkdD DC Al = Bl = Cl = Dl = 90D @D 75o]adltot A I B { lv’glAogoa’j 9v,,u];’pk;gmqjkdaoD K O L 8q;1kj ’G 4hk;jk ABCD c,joI6[lujc9 0hk’0t|koF gIqkwfhG D J C AC = BD c]t OA = OB = OC = OD { ;q’,qomu,j u95f.9dk’ c]t ]aflt\\u OA zkj o9v, A, B,C c]t D D ;q’,qofa’j djk;gxao;q’,qoco[ovdI6[lucj 9lkd ABCD D ສະຫງສ.ວວນ.ລິສຂະສິດ { I6[lulj kd,ulv’cdog7yj’7n IJ c]t KL g-yj’c,joglAodk’lkd0v’0hk’g-y’j |hkdaoF ma’lv’cdo8A’lkddaoc]t 8af16jg,affP;daoda[g,af8afdao0v’lv’ glAogoa’j 9v,D { IJ ⊥ KL F OI = OJ c]t OK = OL #D dkolhk’I6[ 8q;1jk’ !G lhk’I6[lucj 9lkd ABCD 3fp;kj G AB = 8cm c]t BC =5cm A B 8cm y 0=hlq,,5f AB = 8cm c]t BC =5cm D 5cm 0=hlts^5[ lhk’I6[lucj 9lkd ABCD ;ymulhk’G C x { lhk’mjvo-nj AB = 8cm { lhk’g7y’j glAo-nj [Bx) ⊥ ABF;k’g,af C 16j [Bx) 3fp;jk BC = 5cm D { lhk’g7yj’glAo-nj [Cy) ⊥ BC 16jmujg,af C D 128

{ ;k’g,af D 16j [Cy) 3fp;kj CD = 8cm { 0uf8+ D sk A 9twfhI6[lucj 9lkd ABCD mu8j hv’dkolhk’D 8q;1kj ’ @G lhk’I6[lucj 9lkd EFGH 3fp;kj EF =12cm c]t EnFH = 23D xE 12 cm F 0=hlq,,5f EF = 12 cm c]t EnFH = 23D H 23D y 0=hlts^5[ lhk’I6[lucj 9lkd EFGH t G ;ymulhk’G { lhk’,5, xnFy = 23D { ;k’g,af E 16j [Fx) 3fp;jk EF = 12cm D { lhk’,5,lkd FnEt D { [Et) 8af [Fy) 16jg,af H D { 0uflv’mjvodq’j muj,ug,af.9dk’16j H c]t F fh;p]aflt\\u 12cm c]t EH ສະຫງສ.ວວນ.ລິສຂະສິດ 8k,]efa[F lv’mjvodq’j 8afdao16gj ,af G. { 0uf8+ H sk G c]t G sk F 9twfhI6[lujc9lkdmuj8hv’dkolhk’D 8q;1kj ’ #G lhk’I6[lujc9lkd ABCD mu,j ug,af.9dk’16j I F glAogoaj’9v, AC = 10 cm c]t nAIB = 107D xA y B 0=hlq,,5f AC = 10 cm c]t nAIB = 105D 0=hlts^5[ lhk’I6[lucj 9lkd ABCD 107D I mu,j ug,af.9dk’16j I DC ;ymulhk’G { lhk’,5, XnIY = 107D { c8h,;q’,qomu,j u g,af.9dk’16j I fh;p]aflt\\u 4cm 8af[IX) c]t (IY) 16j A c]t B 8k,]efa[ { 0uf8+glAo-nj (AI) c]t (BI) 8af;q’,qo16j C c]t D { 0uf8+ g,af A sk B, B sk C, C sk D c]t D sk A9twfhI6[lujc9lkd muj8hv’dkolhk’D 129

#D [qfg/ydsaf !D 9q’j c8h,I6[lujc9lkd]5j,oU7no c]h;-vdsk7kj 0v’ x c]t y .oc8j]td=]touD x+2 4(x- 1) 2(3 x-4)y = 2x +1 4 (a) (b) 50D 12 − 4x 3yD y − 2 2x +1 (x + 5)D 10x − 2 (d) (c) @D 9qj’lhk’I6[lucj 9lkd ABCD .oc8j]td=]tou]5j,oUD ສະຫງສ.ວວນ.ລິສຂະສິດ dD AB = 7 cm c]t BC = 5cm D 0D AB = 8cm c]t AnCD = 35D 7D AD = 5cm F DB = 7 cm c]t BnDC = 46D ’D BC = 9 cm F AC = 15cm c]t AnCD = 37D #D 9qj’lhk’I6[lujc9lkd EFGH mu,j ug,af.9dk’ I .oc8]j td=]tou8+wxoUD dD EG = 8cm c]t EmIF = 120D 0D EF = 10 cm c]t HnFG = 55D 7D HG = 9 cm c]t EmIF = 126D ’D EF = 8cm c]t IH = 5cm $D .shI6[lk,c9lkd ABC lkd16j A. H,O,K c,jog,afg7yj’dk’0v’mvj o-nj [AB],[BC] c]t [AC] 8k,]efa[D dD 9q’j lhk’g,af D, I c]t L g7y’j 7nda[ A, H c]t K mP[.lj O 8k,]efa[D 0D 9q’j -Uc9’;jk I6[luj ABDC c,joI6[lujc9lkdD 7D 9q’j -Uc9’;jkglAo-nj (HI) c]t (KL) c,jocdog7y’j 7n0v’I6[lujc9lkd ABDC D 130

[qfmu #W I6[fvd9ao c]t I6[9t85]af !D dyf9tde dyf9tdemu ! A la’gdfI6[lujc90hk’0t|ko ABCD 9q’j 0Po8n,j .l[j jvo9Eg,afD DO B { AB = DDDDDDD = DDDDDDD = DDDDDDD { AC ⊥ DDDDDDD faj’oAoF I6[lucj 90hk’C0t|ko ABCD ,uDDDDDDDgmqjkdao c]t DDDDDDDgoaj’9v,8A’ lkddaoS ABCD gvUo;jkG I6[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ 8afg9hp.shgxaoI6[fvd9ao 9kdoAo ra[8k,Ivp0v’lv’glAogoaj’9v, A AC c]t BD D la’gdfgsaoco;.fL { I6[fvd9ao,u9adcdog7y’j 7n c]t ສະຫງສ.ວວນ.ລິສຂະສິດ DO B 9adl6odk’g7y’j 7nL { 9qj’[vdglAomujgxaocdog7yj’7n c]t C g,afmujgxaol6odk’g7yj’7n0v’,aoD dyf9tdemu # B { vy’.loj ypk,0v’I6[lujc90hk’0t |koF I6[lujc9lkd c]t I6[fvd9aoF AI 9q’j oypk, I6[9t85]af8k,I6[lujc9mu1j 60j hk’oUD { 9qj’[vd9eo;ocdog7yj’7n c]t l6o KOL dk’g7y’j 7n0v’,aoD DJ C 131

@D .97;k, !D I6[fvd9ao A dD oypk,G I6[fvd9ao c,joI6[lucj 90hk’0t |kog-y’j ma’lu0j hk’]h;oc8j,u];’pk;gmqjkdaoD D B 8q;1jk’G 4hk;kj ABCD c,joI6[fvd9aoF gIqkwfhG AB = BC = CD = DA C 0D 75o]adltot A { lv’glAogoa’j 9v,8A’lkddao (AC) ⊥ (BD) D 1 2 11 { glAogoaj’9v,gxaoglAoc[j’g7yj’,5,D 2 2O 1 2 B Al1 = Al 2 = Cl1 = Cl 2 c]t Bl1 = Bl 2 = Dl1 = Dl 2 { ,ulv’cdog7yj’7ng-yj’c,jolv’glAogoa’j 9v, C 0v’,aoD (AC) c]t (BD) c,oj lv’cdog7yj’7nD ສະຫງສ.ວວນ.ລິສຂະສິດ { ,ul6odk’g7yj’7n O g-yj’c,jog,af8afdao0v’lv’glAogoaj’9v,D 7D dkolhk’I6[ 8q;1jk’G lhk’I6[fvd9ao EFGH. 3fp;kj ];’pk;0v’0hk’|b’j 0v’ ,aogmqjk 4cm c]t El = 60D. X 0=hlq,,5f EF = FG = GH = HE = 4cm F 4cm c]t El = 60D E 60D 0=hlts^5[ lhk’I6[fvd9ao EFGH. G 4 cm ;ymulhk’G HY { lhk’,5, XnEY = 60D { 0ufmjvodq’j mu,j ug,af.9dk’16j E fh;p]aflt\\u 4cm.mvj odq’j 8af [EX) c]t [EY) 16j F c]t H 8k,]efa[D { 0uflv’mvj odqj’muj,ug,af.9dk’16j F c]t H fh;p]aflt\\u 4cm.lv’mjvodq’j 8afdao16jg,af G. { 0uf8+ Fsk G c]t G sk H 9twfhI6[fvd9aomuj8hv’dkolhk’D 132

@D I6[9t85]af B dD oypk,G I6[9t85]afc,oj I6[lujc9lkdg-yj’ ma’luj0hk’]h;oc8j,u];’pk; gmqjkdaoD AI 8q;1jk’G 4hk;kj ABCD c,oj I6[9t85]afF KO L gIqkwfhG AB = BC = CD = DA D J C 0D 75o]adltot { lv’glAogoaj’9v,8A’lkddao c]t ,u];’pk;gmqjkdaoD (AC) ⊥ (BD) c]t AC = BD F OA = OB = OC = OD { ,ulujcdog7y’j 7ng-yj’c,jolv’glAogoaj’9v, c]t lv’glAodk’lkd 0v’0hk’ g-y’j |hkD (AC) F (BD) (IJ) c]t (KL) c,oj lucj dog7yj’7n0v’I6[9t85]af ABCD. { ,ul6odk’g7y’j 7n16gj ,af8afdao0v’lv’glAogoa’j 9v, c]t lv’glAodk’lkd 0v’ lv’0hk’g-y’j |hkdaoD \\kp7;k,;kj O c,oj l6odk’g7y’j 7n0v’,aoD 7D dkolhk’I6[ ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1jk’G lhk’I6[9t85]af ABCD mu,j u0hk’gmqkj da[ 5cm. x 5 cm C 0=hlq,,5f AB = BC = CD = DA = 5cm B 0=hlts^5[ lhk’I6[9t85]f ABCD. 5 cm 5 cm ;ymulhk’G { lhk’,5,lkd XnAY A 5 cm y D { 0ufmvj odq’j muj,ug,af.9dk’16j A fh;p]aflt\\u 5cm. mvj odqj’8af [Ax) c]t [Ay) 16j B c]t D 8k,]efj a[D { 0uflv’mjvodq’j mu,j ug,af.9dk’16j B c]t D fh;p]aflt|u 5cm. lv’mvj o dq’j 8afdao16gj ,af C. c]h;0uf8+ C sk B c]t C sk D 9twfhI6[mu8j hv’dkoD 133

#D [qfg/ydsaf !D 9qj’lhk’I6[fvd9ao EFGH .oc8]j td=]tou8+wxoUG dD EF = 6cm. 0D HF = 8cm c]t EH = 5cm. 7D HF = 8cm c]t HnEF = 76D. ’D HF = 8cm c]t EnHG = 70D. @D .shlv’g,af A c]t I 16sj kj ’dao 3cm. 9q’j ;k’g,af B, C c]t D grnvj .sh ABCD gxaoI6[fvd9aomuj,ug,af.9dk’16 I c]t 0hk’0v’,aogmqkj 5cm. #D 9q’j lhk’I6[9t85]af WIND .oc8]j td=]tou8+wxoUD dD 0hk’0v’,aogmqkj da[ 7cm. 0D WN =10cm. $D dD 9q’j ;k’lv’g,af A c]t I 9kdoAo9qj’oe.-h[aomaf c]t ;q’;Pogrnvj lhk’g,af B, C c]t D grnjv.sh ABCD gxaoI6[9t85]afmuj,u I gxaog,af.9dk’D 0D 9qj’;k’g,af O c]t lhk’I6[9t85]af RSTU mu,j u O gxaog,af.9dk’ c]t ,u0hk’ 6cm. &D .shI6[9t85]af ABCD mu,j ul6odk’g7yj’7n O. M, N, P, Q c,jog,afg7y’j dk’ 0v’ OA, OB, OC c]t OD 8k,]efa[D ສະຫງສ.ວວນ.ລິສຂະສິດ dD 9qj’-Uc9’;jk MNPQ c,oj I6[9t85]afD 0D 4hk ABCD c,oj I6[fvd9aoD 9qj’[vdxtgrf0v’I6[lujc9 MNPQ. *D .shI6[lujc90hk’0t|ko EFGH Xfaj’I6[?D H G I6h;kj El1 = El 2 c]t Gl1 = Gl2 D 21 9q’j -Uc9’;jk EFGH c,joI6[fvd9aoD 21 F (D .shI6[fvd9ao XYZW Xfaj’I6[?D E Y 9q’j -vdsk7kj 0v’ x .oc8j]td=]tou]5j,oUD dD Xl1 = 3x + 8 c]t Xl 2 = 11x − 24 34 0D Yl3 = 7x −11 c]t Yl 4 = 5x + 9 X 1 5 Z 7D Xl1 = 2x + 7 c]t Zl6 = 3x − 8 2 6 ’D Xl1 = 5(x −1) c]t Zl6 = 2(x +1) 87 9D Yl3 = 4(x + 5) c]t Wl7 = 3(x +15) W 134

[qfmu #! I6[7k’\\6 X0? !D dyf9tde dyf9tdemu ! 9q’j la’gdfg[yj’I6[]5,j oU c]h;[vd-nj0v’c8j]tI6[D { 9qj’[vd95fc8d8kj ’0v’I6[ Xd? da[I6[vnjoMD Xd? X7? X9? X’? dyf9tdemu @ ສະຫງສ.ວວນ.ລິສຂະສິດ xA By .oI6[7k’\\6 ABCD. 0ufglAo-nj (xy) 12 32 zjko9v, A c]t B 0t|ko (CD). { Al1 + Al2 = DDDDDDDDDDDDDDDDDDDDDDDDDDD D C c8;j jk Al1 = Dl. faj’oAo Al2 + Dl = DDDDDDDDDDDDDDDDDDDDDDDD { Bl3 + Bl2 = DDDDDDDDDDDDDDDDDDDDDDDDDDD c8;j jk Bl2 = Cl. fa’j oAo Bl3 + Cl = DDDDDDDDDDDDDDDDDDDDDDDDDD \\kp7;k,;jk .oI6[7k’\\6 ABCD, AB//CD gIqkwfhG Al + Dl = DDDDDDD + DDDDDDD = DDDDDDDD @D .97;k, !D oypk, I6[lucj 9g-y’j 0hk’76j|b’j 0v’,ao0t|kodao gvUo;kj I6[7k’\\6D 8q;1jk’G 4hk;kj ABCD c,oj I6[7k’\\6F gIqkwfhG AB//CD 135

@D xtgrf0v’I6[7k’\\6 dD I6[7k’\\6mq;j wxD { [+,u0hk’.fgmqjkdao AB { [+,u,5,.fgxao,5,lkd 0D I6[7k’\\6mPj ’ DC A B { lv’0hk’mu[j +0t|kodaogmqjkdao { lv’,5,rNogmqkj daoD D C 4hk;kj ABCD c,oj I6[7k’\\6mPj ’ gIqkwfhG E F AD = BC c]t Dl = Cl 7D I6[7k’\\6lkd { ,u|b’j ,5,lkdD H G EFGH c,joI6[7k’\\6lkdF lkd16j H (Hl = 90D) D #D rNoltg]pj rNoltg]pj 0v’I6[7k’\\6gmqjkg7y’j |bj’0v’zqo[;d]ts;jk’lv’rNoD ສະຫງສ.ວວນ.ລິສຂະສິດ ABCD c,oj I6[7k’\\6g-yj’ MN c,oj rNoltg]pj A B 0v’,ao gIqkwfhG MN = AB + CD M N 2 DC 8q;1jk’G lv’rNo0v’I6[7k’\\6 WXYZ cmdwfh 5cm c]t 9cm 8k,]efa[D 9q’j -vdskrNoltg]jp0v’,aoD [qfcdhG WX 0=hlq,,5f WX = 5cm c]t YX = 9cm F M N NM c,jorNoltg]pj 0=hlts^5[ -vdsk NM = L Z Y 9kd0=hlq,,5f gIqk,u MN = WX + YZ 2 = 5+9 = 7 cm 2 136

$D ,5, .o0v’I6[7k’\\6 AB .oI6[7k’\\6 ABCD, AB//CD gIqkwfhG Al + Dl = Bl + Cl = 180D c]t D C Al + Bl + Cl + Dl = 360D 8q;1jk’G 7kj ;afcmd0v’lv’,5,rNo0v’I6[7k’\\6|b’j gmqjk 63D c]t 55D D 9qj’-vdsk7jk;afcmd0v’,5,vnoj D [qfcdhG AB 0=hlq,,5f .sh ABCD c,oj I6[7k’\\6 Dl = 63D c]t Cl = 55D D C 0=hlts^5[ -vdsk Al c]t Bl 9kd0=hlq,,5f gIqk,uG Al + Dl = 180D Bl + Cl = 180D Al = 180D − Dl Bl = 180D − Cl = 180D − 63D = 180D − 55D ສະຫງສ.ວວນ.ລິສຂະສິດ = 117D = 125D &D czo;kf0v’I6[lujc9 I6[lucj 9l;f I6[7k’ I6[lucj 9 lv’760j v’0hk’ 0hk’0t|ko 0t|kodao lv’0hk’ 0t|kodao I6[lucj 9lkd I6[fvd9ao I6[lucj 90hk’0t|ko I6[9t I6[lujc90hk’0t|ko muj,uluj,5,lkd 85]af mu,j ulu0j hk’gmqkj dao I6[lujc9lkd c]t I6[fvd9ao 137

#D [qfg/ydsaf !D 9qj’c8h,I6[7k’\\68k,g’nvj ow0]5,j oU c]t .sh-Uc9’gsfzqo.od=]tou[+lk,kfc8h,D dD ,ulv’,5,lkdD 0D rNoma’lv’lAod;jk0hk’D 7D lv’rNogmqkj daoD ’D ,u,5,cs^,lk,,5, @D 9qj’-vdskrNoltg]jp MN 0v’I6[7k’\\6 ABCD .oc8j]td=]tou8+wxoUD dD AB = 6cm c]t CD = 9cm D 0D AB = 12 cm c]t CD = 8cm. #D 7kj ;afcmd0v’lv’,5,rNo0v’I6[7k’\\6|b’j gmqjkda[ 80D c]t 45D D 9q’j -vdsk7jk ;afcmd0v’lv’,5,vnoj D $D .shI6[7k’\\6 ABCD, MN c,oj rnholtg]pj 0v’,aoD A B dD 4hk CD = 42 c]t MN = 25 c]h; AB = ? 0D 4hk CF = 9 c]h; EN = ? M EN 7D 4hk AB = 5x + 7 y c]t CD = 9x − 3y c]h; MN = ? D FC ’D 4hk ABCD c,oj I6[7k’\\6mPj ’ c]t Dl = 80D c]h; Bl = ? ສະຫງສ.ວວນ.ລິສຂະສິດ &D 9kdI6[F 9qj’-vdsk7kj 0v’ x c]t 7kj c8j]t,5,0v’,aoD AB 135D 9x −13 5x +10 D C *D 9q’j -vdsk7kj 0v’ x 9kdoAo-vdskrNo c]t rNoltg]jpmupj a’[+maoI6hD X0? 2x −1 9 19 Xd? x 15 7x + 3 8− x 3x + 2 X’? 2x + 4 X7? 8 3+ 2x 2x +1 138

[qfmu #@ I6[lk,c9 !D dyf9tde ,uw,hfyh;lk,,af ,af]tlk,fU;D 0t|kfw,hfU;c8j]t,af,ufaj’oUG Xd? 2cm, 3cm c]t 5cm. Xd? X0? 3cm, 4cm c]t 6cm. X0? X7? 3cm, 5cm c]t 9cm. X7? { .oc8j]t,af 9q’j xP[mP[];’pk;0v’lv’fU;da[fU;mujpk;d;kj \\6jD { .oc8]j t,af 9qj’oegvqkw,hfU;,k8+dao.shgxaoI6[lk,c9D oadIPola’gdfgsao co;.fL ,u0=h.fmulj k,kflhk’gxaoI6[lk,c9wfhL c]t ,u0=h.fmu[j +lk,kflhk’gxao I6[lk,c9L phvospa’L { 9kdzqowfhIa[0hk’ gmy’oadIPolk,kflts^5[wfhco;.fdP;da[dkorq;rao]t s;jk’zqo[;d];’pk;0v’lv’0hk’da[0hk’mulj k,0v’I6[lk,c9L @D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ !D 0hk’0v’I6[lk,c9 lk,mvj o-njxtdv[gxaolk,0hk’0v’ A I6[lk,c9|bj’wfhd+8+g,nvj zqo[;d0v’lv’ 0hk’s^kpd;jk0hk’mujlk,D \\kp7;k,;jk .oI6[lk,c9 ABC gIqkwfhG AB + AC > BC c]t AB + BC > AC B C c]t AC + BC > AB 8q;1jk’G gryoj lk,kflhk’I6[lk,c9|b’j mu,j u0hk’gmqkj da[G dD 5cm, 6cm c]t 7cm. 0D 7 cm, 7 cm c]t 14cm. 7D 3cm, 5cm c]t 10cm. wfhs^n[+L phvospa’L [qfcdhG dD lk,kflhk’I6[lk,c9wfh grkt;jk i 5cm + 6cm > 7 cm 139

i 5cm + 7 cm > 6cm 6cm 5cm 7 cm i 6cm + 7 cm > 5cm 0D [+lk,kflhk’gxaoI6[lk,c9 grkt;kj 7 cm + 7 cm =14cm. 7D [+lk,kflhk’gxaoI6[lk,c9 grkt;kj 3cm + 5cm <10cm. @D I6[lk,c9-toyf8jk’M A dD I6[lk,c9mq;j wx { lk,0hk’0v’,ao[+gmqkj daoD { lk,,5,8kj ’daoD B C ABC c,joI6[lk,c9mqj;wxD 0D I6[lk,c9mjP’ A { lv’0hk’gmqkj dao c]tlv’,5,rNogmqkj daoD ABC c,joI6[lk,c9mPj ’ c]t ,u9v,16j A ສະຫງສ.ວວນ.ລິສຂະສິດ gIqkwfh AB = AC c]t ABC = ACB s^n B = C { I6[lk,c9mPj ’,u|bj’cdog7y’j 7nD B D C (AD) c,oj cdog7yj’7n0v’I6[lk,c9mjP’ ABCD 7D I6[lk,c9ltg\\u A { lk,0hk’gmqkj dao c]t lk,,5,gmqjkdaoD ABC c,oj I6[lk,c9ltg\\u gIqkwfhG L M AB = AC = BC c]t A = B = C { I6[lk,c9ltg\\u,ulk,cdog7y’j 7nD BK C (AK), (BM) c]t (CL) c,jolk,cdog7yj’7n0v’ B I6[lk,c9 ABC. ’D I6[lk,c9lkd { I6[lk,c9mu,j u|bj’,5,lkd gvUo;jkI6[lk,c9lkd I6[lk,c9lkd ABC lkd16j A gIqkwfhG A C A = 90 s^n 1D. { 0hk’dq’|hkda[,5,lkd gvUo;kj 0hk’dq’lkdF 140

lj;olv’0hk’8yfcxtda[,5,lkd gvUo;kj B 0hk’,5,lkdD BC c,jo0hk’dq’lkdF AB c]t AC c,jolv’0hk’,5,lkdD { I6[lk,c9lkdmuj,ulv’0hk’,5,lkdgmqkj dao A C gvUo;jk I6[lk,c9lkdmjP’D { lv’,5,cs^,0v’I6[lk,c9lkdmjP’,u7kj ;af cmdgmqjkdaoD B = C #D dkolhk’I6[lk,c9 dD I6h];’pk;0v’ma’lk,0hk’D 8q;1kj ’G lhk’I6[lk,c9 FAT g,njvI6h;kj AF = 5cm, AT = 7cm c]t FT = 9cm. A 0=hlq,,5f AF = 5cm, AT = 7 cm 5 cm c]t FT = 9cm. 7cm 0=hlts^5[ lhk’I6[lk,c9 FAT ສະຫງສ.ວວນ.ລິສຂະສິດ F 9cm ;ymulhk’G T { lhk’mjvo-nj FT = 9cm { 0uflv’mjvodqj’muj,u g,af.9dk’16j F c]t T fh;p]aflt\\u5cm c]t 7cm.lv’mjvodq’j 8afdao16jg,af A. { 0uf8+g,af A sk T c]t A sk F 9twfhI6[lk,c9 FAT mu8j hv’dkolhk’D 0D lhk’I6[lk,c9|bj’g,njvI6hlv’0hk’ c]t ,5,mu1j 6]j ts;kj ’lv’0hk’oAoD 8q;1jk’G lhk’I6[lk,c9 HOT 3fp;kj HT = 5cm, OH = 6cm c]t H = 55 X 0=hlq,,5f HT = 5cm, OH = 6cm c]t H = 55 O 0=hlts^5[ lhk’I6[lk,c9 HOT ;ymulhk’G 6cm { lhk’,5, XHY = 55 55 T { ;k’g,af O c]t T 16j [HX) c]t [HY) H 5 cm Y 8k,]efa[ 3fp;kj HO = 6cm c]t HT = 5cm. { 0uf8+ O sk T 9twfhI6[lk,c9 HOT mu8j hv’dkolhk’D 7D lhk’I6[lk,c9|bj’ g,njvI6h0hk’|bj’ c]t lv’,5,mujcxtda[,aoD 141

8q;1jk’G lhk’I6[lk,c9 WIN 3fp;jk IN = 7 cm, I = 36 c]t N = 60 y x 0=hlq,,5f IN = 7 cm, I = 36 c]t N = 60 W 0=hlts^5[ lhk’I6[lk,c9 WIN ;ymulhk’G I 36 60 N { lhk’mvj o-nj IN = 7 cm 7 cm { lhk’,5, xIN = 36 c]t yNI = 60 { [Ix) 8af [Ny) 16jg,af W gIqkwfhI6[lk,c9 WIN muj8hv’dkolhk’D #D [qfg/ydsaf !D gryjolk,kfc8h,I6[lk,c9muj,u0hk’]5,j ,oUwfhs^n[+L phvospa’L dD 5;6 c]t 10 0D 5;6 c]t 11 7D 5;6 c]t 12 ’D 7;7 c]t 7 @D I6[lk,c9 ABC, AB = 4cm c]t AC = 6cm D 0hk’ BC lk,kfgmqkj da[ dD 11cm F 0D 3cm F 7D 1cm wfh s^n [+L phvospa’L #D 9qj’lhk’I6[lk,c9 ABC g,njvI6h;kj G dD AB = 6cm; AC = 4cm c]t BC = 8cm. 0D AB = 5cm; A = 60 c]t B = 45 ສະຫງສ.ວວນ.ລິສຂະສິດ 7D AB = 5cm; BC = 8cm c]t B = 35 . ’D AC = 7 cm c]t A = C = 80 $D lhk’I6[lk,c9mPj ’ CAT (AC = AT) g,njvI6h;kj G dD CT = 3cm; AC = 5cm. 0D AC = 6cm c]t A = 40 . &D dD 4hk;kj ];’pk;0v’lv’0hk’0v’I6[lk,c9c,oj 15 c]t 20. ];’pk;0v’0hk’mulk, 9t8hv’16j]ts;kj ’ DDDDDDDDDDDDDDDskDDDDDDDDDDDDDDDD 0D 4hk x c,jo];’pk;0v’0hk’mulk,F l;j olv’0hk’vnjoc,jo & c]t (D 9qj’-vdsk7kj 0v’ x. 7D ];’pk;0v’lk,0hk’0v’I6[lk,c9|b’j c,oj x, x + 3 c]t x + 5. 9qj’vdsk7kj 0v’ x. X *D .shI6[lucj 9 ABCD F AC c,oj glAogoaj’9v,0v’,aoD 9qj’-Uc9’;jkG AB + BC + CD + DA > 2AC (D .sh P c,oj g,af.oI6[lk,c9 XYZ D 9qj’ryl6f;jkG P 2(PX + PY + PZ ) > (XY + XZ + YZ ) Y Z 142

[qfmu ## glAomulj e7ao.oI6[lk,c9 ສະຫງສ.ວວນ.ລິສຂະສິດ !D dyf9tde dyf9tdemu ! dD 8afg9hpc0’.shgxaoI6[lk,c9D 0D g9ktI6lk,I616lj k,9v,0v’I6[lk,c9 .shd;hk’r=xt,ko grnvj .sh,ao\\6owx,kwfh 1jk’gl]ug;]kshvp.l/j kD 7D gvqkI6[wxshvp.lj/kmu,j ulkpfyj’ Xfa’j I6[? c]h;0ufglAo8k,lkpfy’j D ’D xt8y[af7n0=h 7 c8jshvp.l9j v,vnoj D oadIPola’gdfgsaoco;.fL { 4hkxt8y[afdyf9tdewfh1jk’4nd8hv’ ma’lk,glAo9t8afdao16gj ,af|bj’D 9D 9q’j gvqkI6[lk,c9fa’j djk;;k’ 1v’.l;j af45.f|b’j s^n oU;,n0v’ 8qogv’ Xfaj’I6[?D oadIPola’gdfgsaoco;.fL lD g,af8afdao0v’lk,glAogvUo;kj spa’L dyf9tdemu @ .shI6[lk,c9 ABC (Al,Bl,Cl gxao,5,cs^,? c]t I6[lk,c9 EDF X Dl gxao,5,s;k?D 9q’j c8h, c]t .sh0=hla’gdfdjP;da[g,af8afdao0v’G dD lk,glAo9v,lkd0v’c8j]tI6[lk,c9D 0D lk,glAo9v,dk’0v’c8j]tI6[lk,c9D 7D lk,glAodk’lkd0v’c8j]tI6[lk,c9D ’D lk,glAoc[j’g7yj’,5,.o0v’c8]j tI6[lk,c9D dyf9tdemu # .shI6[lk,c9 ABC c]t Oc,jog,af8afdao0v’lk,glAodk’lkdD { -Uc9’;kj OA = OB = OC 143

{ c8h,;q’,qomu,j ug,af.9dk’16j Oc]t ]aflt|u OA gIqkla’gdfgsaoco;.f djP;da[9v, A,B c]t C 0v’I6[lk,c9oAoD { ;q’,qo c]t I6[lk,c9 ABC ,udkorq;raodaoco;.fL dyf9tdemu $ A .oI6[c8h,F AI, BI,CI c,oj lk,glAo DE c[j’g7yj’,5,.o0v’I6[lk,c9 ABC D I { -Uc9’;kj ID = IE = IF { c8h,;q’,qomuj,ug,af.9dk’16j I c]t CF B ]aflt\\u r = ID F la’gdfgsaoco;.f dPj ;da[lk,0hk’0v’I6[lk,c9 c]t ;q’,qoL { ;q’,qofa’j djk; c]t I6[lk,c9 ABC ,udkorq;raodaoco;.fL @D .97;k, dD oypk,G glAo-nmj uj0uf9kd9v,0v’I6[lk,ສະຫງສ.ວວນ.ລິສຂະສິດ c9 c]t zkj og,afg7yj’dk’0v’0hk’dq’ !D glAo9v,dk’ |hkgvUo;kj glAo9v,dk’D { I c,oj g,afg7y’j dk’0v’ BC F IB = IC A c]t AI c,oj glAo9v,dk’ muj0uf9kd9v, A D { lk,glAo9v,dk’0v’I6[lk,c98afdao16j CI B g,af|bj’ gvUo;jkG g,af4j;’|ad A XCentre de gravité) 0v’I6[lk,c9D { G c,jog,af4j;’|ad0v’I6[lk,c9 ABC D C' G B' B A' C A 0D ;ymulhk’glAo9v,dk’ { lhk’g,afg7yj’dk’0v’lk,0hk’D BC 144