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")>?@ 4) Phép biªn ð±i không tß½ng ðß½ng :@")>59V:L6E")> ?5%)*")>?@1WIC  4:>5CTAB5BIX5L6 E")>?@)Y%)*")>?@ <Q51< 5" 1.1.5C5$")>?@ :9#Z,:9# . Z[#:;\9#Z,: 4:#] 1x + ; 1x + : 1x 4 1x x 2 − = − ^: 1x 1 1x x − = − _: xx −= 8:#] 2x − ;,] 2x − : 1x + 9# . ]\#: ;` : 2x − 9# . Za#] b:;` c7:c;d±,e\f4:c;d`f:c;d±.f ^:c;∅ _:c;d`f8:c;∅:c;dZ,e`f:c; d.e[f 1.2.5C5$")>?@ :#] 1x 1 − ; 1x 1x2 − − 4:#] 2x 1 − ; 2x 3x2 − − : 2x x − ; 2x 1 − Z 2x −  ^:g#Z.g;#], _: 1x 1x 3x 1x 4x 2 ++ + + = + − 8:g#],g;#Z. :./g#Z,g;#]. :g#Z.g;.#Z , c7:c;d.f4:c;∅:c;∅^:c;d 2 1 f _:c;d\f8:c;∅:c;d`e\f:c; d,f 1.3.5C5")>?@ : x2 x x2 |x| − = − 4: 1x x 1x |x| − = − : 1x 2x 1x |2x| − − = − − ^: 2x x1 2x |1x| − − = − − c7:h`e.:4:9,e]∞::h.e]∞:^:∅ V¤n ð« 2: ijklm no 1) Xét phß½ng trình A.x = B ;`<;p7O<L+")>?@    =∈∀→= φ=→= RS:Rx0x.0 S:VNconstx0 ≠`<≠p75%)*#; A B  !"51<c;d A B f 2) Các phß½ng trình ðßa v« b§c nh¤t khác(Gq%r%5sIF5151<L+ "C5#J%52IF51!51< 5"  2.1.5C5L+451'I! :9< . ].:#Z.<;#Z[ 4:<9#Z<:;#]<Z. :<9#Z<][:;<9#Z.:]b ^:< . 9#Z,:]<;#9[<Z.: _:9<],: . #;9.#],:<]a#]. 8:9#].4 . :Z [ ;4 . 9#]: :9]4: . #]. . ;.9]4:]9 . ]4 . :# :9#].4 . :Z . ;4 . 9#]: 2.2.5C5L+451'I! : m 3x 2mx)1m( = + −++ 4: 1 1x 3mmx = + −− tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y ,    : 2 x 3x 2x mx = + + − − ^: 2 mx 1x 1x mx = − − + − − _: 2 x1 2)1x(b x1 bx 1x xa − −− = + − + − − 8: ax bx ax bax + − = − + 2.3.5C5L+451'I! :g#]<g;g#Z<].g 4:g#Z<g;g#] ,g :g<#],g;g.#]<Z[g ^:g<#],g;g[#] <Z.g 2.4.5C5L+451'I! : 2x 3x 1x mx − + = − + 4: 3 mx 2mmx = − −+ : 1x )1x(a 1x b 1x 1ax 2 2 − + = + + − − ^:9#Z<:9<#Z[: ;` V¤n ð« 3: mzk{ jk0|jk0|cjk0 1)Ði«u k¸ên Ax = B có nghi®m duy nh¤t    ≠ F5}+5}@I~5}< 0A 2) Ði«u ki®n Ax = B nh§n ∀x∈  là nghi®m    = = 0B 0A 3) Ði«u ki®n Ax = B có nghi®m  •J 51<^IOQ  •J 51<'+∀#∈ 4) Ði«u ki®n Ax = B vô nghi®m  ;`L+≠`   51<L+4&'M5 5" 3.1.@<<%P")>?@ 51<^IO Q :9<],: . #],Z<;9y<Za:# 4: mx 2x 1x 1x − + = − + : 2 x 2x 1x mx = − + + + c7:<≠.-[4:<≠Z.-`-,:<≠[-, 3.2.•$%&<%P")>?@ !" 51<'+ :< . #;v#]< . Z\<][ 4:< [ #;< . ] <#Z< :9#Z,:]9.#],:4;#]. ^:9.#Za:]9\Z[#:4]bZ#;` c7:[4:`-,:;Z,-4;,^:; .-4;, 3.3.@<<%P")>?@LU51< : 2 x 2x 1x mx = − + + + 4: 1x 3m2x 1x4 1x mx2 − +− =−− − + :9#Z,:< . ]#<].#Z,;` ^: 2 mx 1x 1x mx = − − + − − c7:<;,-[4:<≤.u[:∅^:<≠, 3.4.@<<%P")>?@ 51< :.9g#g]<Z,:;g#gZ<][ 4:< . 9#Z,:;\#Z[<].LK5#€` : 2x 1m2x2 2x 2x mx3 − −+ =−+ − − ^: 2x − 9<#Z<]\:;` c7<≤[ua4:<•Z.L<€,:<€,^: I‚q V¤n ð« 4: k •J1")>?@    =+ =+ 222 111 cybxa cybxa  w; 22 11 ba ba ; , /4 . Z . /4 , / w # ; 22 11 bc bc ; , 4 . Z . 4 , / w O ; 22 11 ca ca ; ,  . Z .  , / 6Iw≠`<≠p 51<^IO Q'+ r"AB9#; D D x -O; D D y :  6Iw;`L+w # ≠`Lw O ≠`71LU 51<  6Iw;w # ;w O ;`71 LUAB 51<C , #]4 , O; , 5" 4.1.5C5$1")>?@ tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y .    :    =+ =+ 1,1y7x2 8,7y24x27 4:    =− =+ 21y5x13 23y9x7 :    =− =− 8y9x7 3y4x5 ^:    =− =+ 11y5/3x2/5 16y3/2x4/3 _:      =+ =− 3y2x5 1yx3 8:      =−− −=++ 22y)12(x2 12yx)12( 4.2.1    =− =− 154v7u3 50v9u5 /~O5C51L+AIO ?51<E1        = − + + = − − + 308 2y 7 3x 3 100 2y 9 3x 5  c7I;.ƒ-L;,`#;Z,byuab-O;\,u.` 4.3.5C5L+451'I!pF51 51<@< 135„#-O%='!"%B5LK5<r- 4 :    +=+ +=+ 5m2myx2 1my2mx  4:    =+++ =−+ 2y)1m(x)2m( 5y)2m(mx :    −=−+ −=+− m1yx)2m( 1m3y2x)1m( ^:    =+ +=−+ 2myx2 1my)1m(mx _:    =−+− =+−+ my)4m(x)1m2( 4y)2m(x)4m( 8:      =− =− 2 2 bybx ayax :    =+ +=+ ab2aybx babyax 22 :      =− −=− b4ybbx babyax 2 2 V¤n ð« 5: mzjkk {jk0-|jk0-{|cjk0 1) H® có nghi®m duy nh¤tF5w≠` 2) H® vô nghi®m :  …w-5C5w;`< O<L+15")>?@F$ I@LU51< 3) H® vô s¯ nghi®m : …w-5C5w;`< O<L+15")>?@5B I@ LUAB51< 5" 5.1.1    +=+ =+ 1m2ymx m3myx 9<<AB: :&<%P1 51<^IOQp56 13'5'M5„$51<%='!"%B5 LK5< 4:@<<∈%P51<E1'+$AB IO c7:<≠±,-9#9#Z.:;9OZ,:9OZ[:4:Z[- Z.-Z,-` 5.2.1    +=+ =+ 1mmyx m2ymx 9<'+<AB: :&<%P1LU51<p 4:&ABIO<%P1 51<^IO Q'+ABIOp…$51<% p c7:<;Z,4:<;`79,e`:-<;Z.79[e.: 5.3.&<%P1 &<^IOQ :    =−+++ =−++ 0m31y)3m(mx 0m4y8x)1m( c7<≠,-<≠[ 4:        −=+− =++ )1m(2 y 2 x 2 )2m( m y 1 m x 2 )1m( c7<≠`-.-.± b -Z,± 3 5.4.@<%52IF51%P1 LUAB51< tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y [    :    +=++ +=+− m3y2x)6m( m1myx4 c7<;Z. 4:    −=+=+ −=+++ 1by)ba(2x)a5( aby)ba(x)a1(  c7;[-4;aL;Z4;Z\± 17 5.5.@<%52IF51-4%P1LU51<    −=+ +=+ baaybx babyax c7;±4≠` V¤n ð« 6: †0‡† ˆj‰oŠi 1)иnh nghîa7≥4Z4≥` ‹$?)Y*"^)>7 . ≥`-gg≥`- A ≥`-…E5ABH^QI/≥`- Œ$4@")> . ] . ] . ≥` 2) Tính ch¤t 7 :…=7 ‹=5L6H<=AB7 ≥]F≥]F ‹=LLKL6E54Q%‡3H 52I7 ≥n≥w]≥]w 9Gq7FU%)*?TL6LKL6: 4:…7 ‹5L6E4Q%‡3H <=AB^)>%)*4Q%‡3H52I -AB<@4Q%‡3%Œ52I ‹≥LK/€` A 1 ≤ B 1 ‹≥€`  ≥  /n n A ≥ n B :…4•(I7 ≥n≥≥ 5" 6.1.wH%&Ž3<57 :∀-4-∈7 . ]4 . ] . ≥4]4] 4:∀-4∈74]]4≤,] . ]4 . / :∀-4--^-_∈7   . ]4 . ] . ]^ . ]_ . ≥94]]^]_: ^:∀-4-∈7   . 9,]4 . :]4 . 9,] . :] . 9,] . :≥b4 6.2.3<5 :∀-4∈ ] 7 2 a b b a ≥+ 4:∀-4∈7 2 22 2 ba 2 ba       + ≥ + :]4€`7 3 33 2 ba 2 ba       + ≥ + ^:∀≠`7 2 ≥+   ,  6.3.∀-4-#-O∈/3<54Q%‡ 3 I5AU"AF57g/#]4/Og≤ )yx)(ba( 2222 ++ T% AIO?7a/A5],.A≤,[ 6.4 3<5$4Q%‡3?4)A_" :6I≥4-#≥O7.9/#]4/O:≥9]4:9#] O: 4:6I≥4-#≤O7.9/#]4/O:≤9]4:9#] O: :6I≥4≥-#≥O≥•7 [9/#]4/O]/•:≥9]4]:9#]O]•: ^:6I≥4≥-#≤O≤•7 [9/#]4/O]/•:≤9]4]:9#]O]•: _:T% AIO?7?<D∆'IU'IU    5:]4]≥ 3 π 9]4]:  55: ≤ ++ cba hhh S.18 ]4] 6.5.]4;,/•$<5 : . ]4 . ≥ 2 1  4: [ ]4 [ ≥ 4 1 : \ ]4 \ ≥ 8 1 w7: . ]9Z4: . ≥,]`4:%PqZ4≥Z 2 1 9 . ]4 . : :9 . ]4 . : . ]9 . Z4 . : . ≥ 4 1 ]` 6.6.-4-'+4M<5$ :/4/≥9]4Z:9Z4]:9Z]4]: 4: cba 3 ac 1 cb 1 ba 1 ++ > + + + + + : [ ]4 [ ] [ ≥[//4/ w7:≥g4Zg . ≥ . Z94Z: . -… 4:]4]€]4-…=:9///:/≥` 6.7.3<5 :6Igg≤,-g4g≤,@g]4g≤g4],g 4:6I-4-<€`L+ b a ≤,@ b a ≤ mb ma + + tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y \    :6I-4-<€`L+ b a ≥,@ b a ≥ mb ma + + w7:4@5L64:L@≤4:L@≥4 6.8.3<5 :6I≥4≥,@ b.a1 2 b1 1 a1 1 22 + ≥ + + + 4:6I#-O∈L+#]O€`@ yxyx 21 2 41 1 41 1 + + ≥ + + + w7:WIO%‘%)L29Z4: . 94Z,:≥`4: )>’ 6.9.-4--^-#-O-•€`/3<5 :,• ac c cb b ba a + + + + + •. 4:.• bad ad adc dc dcb cb cba ba ++ + + ++ + + ++ + + ++ + •[ :6I z c y b x a ≤≤ @ z c z c y b x a x a ≤++≤ w7:u9]4]:•u9]4:4:9]4:u9]4 ]]^::•#9u•:L+;#9u#:=L6 F6WIC 6.10.3<5 :6I-4--^€`L+ 22 db cdab d c b a + + << 4  @5 4:6I-4--^€`@ db 1 ca 1 d 1 c 1 b 1 a 1 111 ++ + ≤ + + + :6I€4€`L+<€-<-∈@   nn nn mm mm ba ba ba ba + − > + − ^:6I 3 4 b1 1 a1 1 ≥ + + + @/4≤ 4 1 w7:WIO%U^Z4≥`9%:4:WIO%‘ 9^Z4: . ≥`9%::WI5%‘ <Z €4 <Z  9%: V¤n ð« 7: †0‡† ˆQ%‡3IO 1) B¤t ðÆng thÑc Cauchy cho 2 s¯7   -4≥`7  ]4≥. ab 4≤ 2 2 ba       +  wQI%‡3#CO?F5;4 2) B¤t ðÆng thÑc Cauchy cho n s¯ :  AB , - . -////-  ≥`  , ] . ]////]  ≥ n n21 a aa  ,  . /////  ≤ n n21 n a .aa       ++  wQI%‡3#CO?F5 , ; . ;///;   / 3) H® quä : ‹6I5AB-4 ŒFU%Œ@… %MlF5;47/4≤ 2 2 ba       + ‹6I5AB-4 …FU%Œ@Œ %MF5;47]4≥. b.a 5" 7.1.wH4Q%‡3IO5AB 3<57 :-4€`7 a b b a + ≥. 4:-4≥`79]4:9,]4:≥\4 :-4€`7       +       + a b 1 .b a 1 ≥\ ^:-4€`7       +       + a b 1. b a 1 ≥\ _:-4€`7       +       + b b 1 a a 1 ≥\ 8:-4-€`7 8c c 1 b b 1 a a 1 ≥       +       +       + :-4-€`7 81 a c 1 c b 1 b a ≥       +       +       + :-4-€`7 8a c 1 c b 1 .b a 1 ≥       +       +       + 5:-4€`7 1k kk 2 a b 1 b a 1 + ≥       ++       + eF∈ 7.2.-4≥,/3<5 : 2 ab 1b.a ≤−  4: b.a1ab1b.a ≤−+− : cbacabcb.a ++≤++ 7.3.-4-≥`/cN^“4Q%‡3 IO5AB3<54Q%‡3 IO4AB7]4]≥[ 3 c.b.a 7.4.-4--^€`/3<5 :9]4]:/9 c 1 b 1 a 1 ++ :≥v 4:4]4]≥[ 3 222 c.b.a :9,]:9,]4:9,]:≥9,] 3 c.b.a : [  ^: 9 4 3 abc.9c4b3a2 ≥++ _: b ]4 v ≥\9 . 4 [ Z,b: 8: . 4]4 . ] . ≥[4 tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y a    : )cb)(da()db)(ca()dc)(ba( ++++++++ ≥ ≥b/ 4 abcd : cdab)db)(ca( +≥++ 7.5.-4-€`/07 : . 4]4 . ] . ≥49]4]: 4: [ ]4 [ ]4≥49]4]: : abc 1 abcca 1 abccb 1 abcba 1 333333 ≤ ++ + ++ + ++ 7.6.-4--"'+4M<5$-"'+ NIL5E<5$/07 : ba 4 b 1 a 1 + ≥+ 4:       ++≥ − + − + − c 1 b 1 a 1 2 cp 1 bp 1 ap 1 7.7.-4--^€`/07 : 2 3 ca b cb a ba c ≥ + + + + + 4:6I/4//^;,@9,]:9,]4:9,]:9, ]^:≥,b :6I]4];,@7 5 3 c2 c b2 b a2 a ≥ − + − + − ^:6I]4];,@ 4 3 c1 c b1 b a1 a ≤ + + + + + _:6I 2 c1 1 b1 1 a1 1 = + + + + + @/4/≤ 4 1 8:6I]4];,@ 64 c 1 1. b 1 1. a 1 1 ≥       +       +       + V¤n ð« 8: w”‡†0ojl• oj–—0c˜   GHI CHUÁ 7  ,: 7/4≤ 2 2 ba       + ;™AB  94: <# ;™ABF5;4 .: 7]4≥. b.a ;™AB  9]4: <5 ;™ABF5;4 ZZZZZZZZZZZZZZZZZZZZ 8.1.@<5$?&'KQE+<ABL+?&AB E#)>3p :O;89#:;9.#Z,:9[Z#:LK 2 1 ≤#≤[ 4:O; x2 4x − LK#∈ :O;#9[Z 3 #:LK`≤#≤ 3  ^:O; x3 2x − ],F5#≥. c7:.auƒ-#;yu\4:,uƒ-#;ƒ:[ 3 u\-#; 3 u.^:,ub-#;\ 8.2.@<E+<ABAI :O; 1x 2 2 x − + F5#€, 4:O;[#] 1x 4 + F5#€Z, :O; x 1x 3 + + F5#≥` ^:O; 1x 1xx 2 + ++ F5#€Z, _:O; 3x 5x3x2 2 + +− F5#€Z[ 8.3. :@<l7O;9.Z[#:9.#][:F5Z 2 3 ≤#≤ 3 2 4:@<7O; 1x 4x2x 2 − ++ F5#€, V¤n ð« 9: nk  ,:Giäi và bi®n lu§n b¤t phß½ng trình7 ?)Y *" /#≥ /#≤ €` #≥ A B #≤ A B •` #≤ A B #≥ A B ;` `/#≥AB < ∀#∈ `/#≥AB ^)> ∅ `/#≤AB < ∅ `/#≤AB ^)> ∀#∈ .:D¤u cüa nh¸ thÑc b§c nh¤t f(x) = a.x + b  # Z∞Z a b  ]∞ 89#: ?$5^QILK1AB`H^QILK 1AB 5" 9.1.5C5L+451'I! 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≥−++−++−+ 11.2.3<56I-4€`L+]4;, @7 2 25 b 1 b a 1 a 22 ≥       ++       + w7# . ]O . ≥ 2 1 9# . ]O . :L+ 4 ab 1 ≥ 11.3.€4€`/3<57 9 [ ]4 [ :9 a 1 ] b 1 :€9]4: . / w79]4: . ;9 b 1 b a 1 a 33 + : . / 11.4.# . ]O . ;,/3<5 :,]4]9]4:#]94Z,:O≤,]4] 22 )b1()a1( ++ 4: tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt ,au.vece"w5Ae5055Ox</L/L-x8"/L/L vvb[a`y ƒ . E")>?@-5C5")>?@'+%5@< $?&AB#  C<1%289#:;9#:/!" *"S$?&AB#  '+!"*"51<E ")>?@. 51<L+4&'M5 5" 3.1.@<<%P")>?@ 51<^IO Q :9<],: . #],Z<;9y<Za:#

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