Tuyển Chọn 50 Bt Bpt

7/23/2019 Tuyn Chn 50 Bt Bpt

1/21

TUY N CHN50 BI TON

GII BT PHNG TRNH

CM NANG CHO MA THI

NGUY N H U BI N

https://www.facebook.com/groups/nguyenhuubienEmail: [email protected]

(N THI THPT QUC GIA)


2/21

TUYN CHN 50 BI TON GII BT PHNG TRNH - N THI THPT QUC GIA

Trang 1NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien

Bi 1:Gii bt phng trnh 2 21 2 3 4 .x x x x+ Hng dn

- iu kin: 2

2

0 0 13 41

1 0 0 .3 41 3 41 82 3 4 0 8 8

x x

x xx

x x

+

+

- Bt phng trnh cho tng ng vi2 2 21 2 (1 ) 2 3 4x x x x x x+ + 2 23( ) (1 ) 2 ( )(1 ) 0x x x x x x + + +

2 2 22

5 341 93 2 1 0 9 10 1 0

1 1 1 3 5 34.

9

xx x x x x x

x xx x x

x

+

+ + + + +

- Kt hp iu kin (*), ta suy ra nghim ca bt phng trnh l5 34 3 41

.9 8

x + +

Bi 2:Gii bt phng trnh )(,01102492321 22 Rxxxxxx +++

Hng dn: iu kin: 1x - Bt phng trnh cho tng ng vi

0410249423211 22 ++++ xxxxx

[ ]

)1(03)13(223

6

11

1)2(

03)13()2(223

)63(2

11

2

0)269)(2)(223(2)11(

2

2

2

+

++

+

++

+

+

+

xxx

x

xxx

x

x

x

xxxxx

- Dthy ( ) 1,013)11.3(313223

6

11

1 22>=>+

++

+ xxxx - Hn na (1) .202 xx Kt hp iu kin thu c .2x

Bi 3:Gii bt phng trnh sau: ( ) ( )2 2 21 log log 2 log 6x x x+ + + >

Hng dn:K: 0 6x< < .

( ) ( )22

2 2log 2 4 log 6x x x + > ( )22 22 4 6 16 36 0x x x x x + > + >

Vy: 18x< hay 2 x< So snh vi iu kin. KL: Nghim BPT l 2 6x< < .

Bi 4:Gii bt phng trnh )(,1422

7119229

23

23

Rx

xxx

xxxx>

++

++

Hng dn:iu kin

++

0422

123 xxx

x

- Nhn xt 1,014221422 23 >=++++ xxxx .- Bt phng trnh cho tng ng vi

0217248114227119229 232323 >++++>+ xxxxxxxxxxx


3/21



)1(01)12(211

1)2(0)188)(2(

11

2 22>

+

+>++

+

x

xxxxx

x

x

- R rng 1,011)12(21)12(211

1 22>=>+

+xx

xnn (1) 202 >> xx

Bi 5:Gii bt phng trnh: ( ) ( ) ( )5 5 15

log 4 1 log 7 2 1 log 3 2x x x+ + +

Hng dn:+ iu kin: 1 74 2

x < <

( ) ( ) ( )5 5 5log 4 1 log 3 2 1 log 7 2x x x + + + +

( )( ) ( )

( )( ) ( )

5 5

2

log 4 1 3 2 log 5 7 2

4 1 3 2 5 7 2

12 21 33 0

331

12

x x x

x x x

x x

x

+ +

+ +

+

Giao vi iu kin, ta c:1

14 x < . Vy: nghim ca BPT cho l1

14 x <

Bi 6:Gii bt phng trnh )(221452)1( 22 Rxxxxxxx ++++

Hng dn: iu kin: .Rx Khi :

0)5212(2)522)(1( 222 ++++++ xxxxxxx

05212

547)52)(1(252214)1(

0)5212

)13(2522)(1(

05212

)13)(1(2)522)(1(

05212

)5244(2)522)(1(

22

22222

22

2

22

2

22

222

+++

+++++++++

+++

++++

+++

+++++

+++

++++++

xxx

xxxxxxxxx

xxx

xxxxx

xxx

xxxxxx

xxx

xxxxxxx

- Do >++=+ 16)2(547 222 xxxx 0 nn (2) )1;(101 + xxx

Bi 7:Gii bt phng trnh : ( ) 2 2x 1 x 5 x x 1 + + > +

Hng dn:x 1+ : loi

( )

22 2 2

2 2

2

2

x x 1 1 1x 1: x 5 x 5 x x 5 xx 1 x 1 x 1

5 15 x 1 x 5 x 4x 5 x 5

x 1x 5 x

5x

x 24

15x 40x 20 0

++ > + > + > + + >

> > + + > ++ +

>

> + >


4/21



Bi 8:Gii bt phng trnh: ( )2 25 4 1 ( 2 4)x x x x x+ < + + (xR).

Hng dn: ( )2 25 4 1 ( 2 4)x x x x x+ < + + (*)

- K: x(x2+ 2x 4) 0 1 5 0

1 5

x

x

+

- (*) 2 24 ( 2 4) 5 4x x x x x+ > + 2 24 ( 2 4) ( 2 4) 3x x x x x x+ > + + (**)

TH 1: 1 5x + , chia hai vcho x > 0, ta c: (**) 2 22 4 2 4

4 3x x x x

x x

+ + > +

t2 2 4

, 0x x

t tx

+ = , ta c bpt: 2 4 3 0t t + < 1 3t < <

22

2

7 4 02 41 3

4 0

x xx x

x x x

1 17 7 65

2 2x

+ +< <

TH 2: 1 5 0x , 2 5 4 0x x+ < , (**) lun tha mn

Vy tp nghim BPT (*) l1 17 7 65

1 5;0 ;2 2

S + + =

Bi 9:Gii bt phng trnh sau : 2 5 3 2 4 1 5 6x x x x+ + > + + Hng dn:

2 5 4 1 3 2 5 6 0

1 1( 2 4)[ ] 0

2 5 4 1 3 2 5 6

2

BPT x x x x

xx x x x

x

+ + + >

+ + >+ + + +

- Do (*) 2 0 2x x , kt hp vi iu kin5

2x ta suy ra bt phng

trnh cho c nghim l5

22

x

Bi 11:Gii bt phng trnh )(76)1(2152

)2(2 2 Rxxxx

x++++

++

+

Hng dn:iu kin:2

5x

Bt phng trnh cho tng ng vi

)1(0)3(2652

1)1(0)3)(1(2

652

1

0)32(265276242152 22

++

+++++

+++

++++++++++

xxx

xxxxx

x

xxxxxxxx

Ch rng2

5,0)3(2

552

1>++

+++

xxxx

nn (1) 101 xx

Vy bt phng trnh cho c nghim 1x

Bi 12:Gii bt phng trnh2 8

2 1 2x xx x

+

Hng dn:iu kin ca bt phng trnh:

221 0 0 2 0

8 222 0

2 0

x

x xx

xxx

x x

<


6/21



( )2

3 2 3 22( 2 4 8) 4 0 2( 2 4 8) 4x x x x x x + + =

3 2

0

2 4 0 1 5 1 5

1 5

x

x x x x x

x

=

= = + = +

=

(do 2x )

Vy bt phng trnh cho c tp nghim l [ ) { }2;0 1 5 + Bi 13:Gii bt phng trnh sau : 22 1

2

log ( 1) log ( 1)x x .

Hng dn: K: x >1. BPT2 2

2 1 2 22

log ( 1) log ( 1) log ( 1) log ( 1) 0x x x x +

2( 1)( 1) 1x x

3 2 1 1x x x + 2 ( 1) 0x x x 1 5

2x +

(dox >1).

Vy tp nghim ca BPT l1 5

S= ;2

++

.

Bi 14:Gii bt phng trnh 3 32log ( 1) log (2 1) 2x x +

Hng dn:K: 1x > . BPT 12

33

2log ( 1) log (2 1) 2x x +

3 3 3log ( 1) log (2 1) 1 log ( 1)(2 1) 1x x x x +

2( 1)(2 1) 3 2 3 2 0x x x x

1 22

x . Kt hp K ta c tp nghim l ( ]1;2S =

Bi 15:Gii bt phng trnh )(,)1()12)(3( 22 Rxxxxx +

Hng dn:iu kin:2

1x

- Nhn xt x = 1 khng tha mn bi ton, do xx 12 - Bt phng trnh cho tng ng vi

2133

,2

13301312212

22133)12(3)12(

)1(3

2222

22

2

2

++++

+

xxxxxxxxxxx

xxxxxxxxx

xx

Kt hp iu kin ta thu c nghim2

313 +x

Bi 16:Gii bt phng trnh 29122)5124(4 2223 ++ xxxxxxx

Hng dn: +) iu kin:

0

2022

x

xxx


7/21



+) Ta c bt phng trnh cho tng ng vi

[ ] )1(0)()12(02)52(252)12(02)52)(12()252)(12(

02)5124(29124

22

23

2223

+

+

++

xfxxxxxxx

xxxxxxx

xxxxxxx

+) Vi xxxxxxf 2)52(252)( 22 += .t xxttxxt 2)0(;2 222 ==

- Khi 2)52(22)52()2(22)52(252 2222 +=+=+ xtxtxtxxxxxxxx

- Ta c 2222 )32(912416825204)2(8)52( =+=++== xxxxxxxx

Do vy phng trnh

=

=

=

2

1

20)(

t

xt

xf

Do vy ta c phn tch

122)(22(2)52(252)( 2222 ++=+= xxxxxxxxxxxf

Khi (1) 0)122)(22)(12( 22 ++ xxxxxx

)2(,0)22)(12( 2 + xxxx

(Do 2 0122 >+ xx vi mi x thuc min xc nh)Ta xt mt strng hp sau:

+) TH1:2

1012 == xx (khng tha mn)

+) TH2) 2442

222

22

2=

+=

= x

xxxx

xxxx (tha mn)

+) TH3

+

442

2

22

01222

2 xxxx

x

xxx

xHphng trnh v nghim

+) TH42

1

22

012

2


8/21



Giao vi iu kin, ta c:1

14

x < . Vy: nghim ca BPT cho l1

14

x <

Bi 18:Gii bt phng trnh: 2 2(4 7) 2 10 4 8x x x x x + > +

Hng dn:K: x -22 2(4 7) 2 10 4 8x x x x x + > + 2 2(4 7) 2 2(4 7) 2[( 2) 4]x x x x x x + + > +

2

(4 7)( 2 2) 2( 2 2)( 2 2)x x x x x + + > + + +

2 2

2 2

4 7 2 2 4 4 2 2 2 1

(2 ) ( 2 1) 0 (2 2 1)(2 2 1) 0

x x x x x x

x x x x x x

> + > + + + +

+ + > + + + + >

2 2 1

2 2 1

x x

x x

+ >

+ < hoc

2 2 1

2 2 1

x x

x x

+ >

+ <

Gii cc hbt pt trn c tp nghim l: T = [ )5 41

2; 1 ;8

+ +

Bi 19:Gii bt phng trnh 38 2 (4 1)( 14 8 1)x x x x x + + + .

Hng dn:iu kin : 1x 3 3 3(1) 8 2 (4 1)( 1 8 1 16 1) 8 2 (4 1) (4 1) (2)x x x x x x x x x + + + + +

- Xt hm s 3 2( ) ; '( ) 3 1 0 1f t t t f t t t= = > f(t) ng bin trn [1;+ ) m (2) c

(2 ) (4 1)f x f x + v 2 ,4 1 [1; )x x+ + nn (2) 2 4 1x x +

2

2 4 0

2 4 1 (2 4) 1

1 0

x

x x x x

x

2

22 17 17

17 17 17 17 84 x 17 x 17 0 ;8 8

xx

xx x

+

+ +

Bi 20:Gii bt phng trnh: ( ) 2( 2) 2 3 2 1 2 5 3 1x x x x x + + + + + + Hng dn:iu kin: 1x

t

2 2

2

2 2

22 3

1 2 5 3, 0 1 2

x a bx a

x b x x aba b a b

+ = + =

+ = + + = =

.

Bt phng trnh trthnh: 2 2 2 2( )( 2 ) 2a b a b ab a b + 2 2 2 2( )( 2 ) ( ) ( ) 0

( )( 2 ) ( 2 ) 0 ( 0)

( 2 )( 1) 0

a b a b b a b a b

a b a b a b do a b

a b a b

+ +

+ >


9/21



TH1:

111 1

2 3 2 1 0 32 2

2 3 1 1 0 1 3

xx

x x x x

x x x

+ + + +

TH2:

11

12 3 2 1 0 12

2 3 1 1 0 1; 3

xx

x x x x

x x x x

+ + =

+ +

Vy bt phng trnh c nghim1

{ 1} ; 32

S =

Bi 21:Gii bt phng trnh 5325235010 22 + xxxxx

Hng dn:iu kin

10

74525

5

0252

0350102

2

+

+

x

x

xx

xx

- Nhn xt 053252

4714253252

2

22

>

++

+=+

xxx

xxxxx

- Bt phng trnh cho tng ng vi

02.51123)2(5)5112(2

02.)5)(12(320274

)5)(2)(12(645925235010

22

2

22

+++

++

++

xxxxxx

xxxxx

xxxxxxxx

- t )0;0(,2;5112 2 >>==+ babxaxx ta thu c

2

226;

2

2260712225112

0)52)((0352

22

22

+++

++

xxxxxxx

bababaabba

Kt hp iu kin ta c tp nghim

++= ;

2

223S

Bi 22:Gii bt phng trnh xxxxx 215123 232 ++

Hng dn:iu kin 20)2(

1

05123 2

+

x

xx

x

xx


)1()1)(1(2125123 2232 +++++ xxxxxxxxxx

0.232)23(3)(

0)1(.2)(1(26102

232223

223

+++++++

++++

xxxxxxxxxx

xxxxxxxx


10/21



)1(023

223

.3123

2

23

2

++

++

++

+

xxx

xx

xxx

xx

t )0(23

23

2

=++

+tt

xxx

xxth (1)

)2(0242313

10231 32322 ++++++ xxxxxxxttt

Nhn thy (2) nghim ng vi 2x . Kt lun nghim [ )+= ;2S

Bi 23: Gii bt phng trnh:23 4

2 2 311

x x x

xx

+ + ++ +

++

Hng dn:K: x > -1

- Theo cu a ta c:2 4

3, 11

+ + >

+

x xx

x. (1)

- Li c3 2

11 1

+= + +

+ +

xx

x x

- p dng bt ng thc C si cho hai s 21,1

++

xx

ta c:

21 2 2, 1

1+ + >

+x x

x (2)

T(1) v (2), cng vvi vta c:23 4

2 2 311

x x x

xx

+ + ++ +

++, 1x >

Suy ra mi gi trx > -1 u tha mn bt phng trnh.Vy kt hp vi iu kin, bt phng trnh c tp nghim l ( )1;S = +

Bi 24:Gii bt phng trnh sau:

2

21 2 2 3 1 11 2 1x x x

x x

+ + +>

+

Hng dn:iu kin: 2

2

0

3 1 0 0

1 2 1 0

x

x x x

x x

+ +

+

- Ta c2

2 1 32 1 2 3 1 ( 0)2 4

x x x x

+ = + >

21 2 1 0x x + <

- BPT2 21 3 1 + + < + +x x x x x

1 11 1 3x x

x x + + < + + (V x = 0 khng tha mn bt phng trnh)

- t1

2x t tx

+ = v 0x > .

- Ta c13

1 1 3 2 1 34

t t t t + < + < <


11/21



- Suy ra13 1 13

2 24 4

t xx

< + <

( )2

2

12

1 0 13 105 13 105

1 13 8 84 13 4 04

xxx

xx xx

x

+ +

<

>

>x

x

x

xx

xx

xxx

xxx

ba

ba(do )1x

- TH2: 3113131085

022

84

2

2 2

2

2

2

+


12/21



Bi 27: Gii bt phng trnh 2.14 3.49 4 0x x x+

Hng dn:Chia chai vca bpt cho 4xc bpt2

7 72 3 1 0

2 2

x x

+

t7

2

x

t =

(vi t > 0). BPT trthnh 3t2+ 2t 1 0

11

13

3

t

tt

7 1

2 3

x

7

2

log 3x . KL: BPT c tp nghim

+= ;3log

2

7S

Bi 28:Gii bt phng trnh )(4307545124 23 Rxxxxxx + +

( )0x .

( )2

( 2) | 2 | 1 1 2 1x x x x x + + > + +

. (1)

* 2 :x = (1) 0 2 2 > (loi).* 0 : (1) 2 2= > x (loi).


13/21


14/21



T (1) v (2) suy ra ( ) 0 0g x x> > .+ ( ) 0 4 0 4f x x x> > > . Kt hp K suy ra p s: 4x > .

Bi 31:Gii bt phng trnh )(1928 233 Rxxxxx ++

Hng dn:iu kin: 30)3)(3(

2

01

092

08

2

23

3

++

+

xxxx

x

x

xx

x


2332

3320)332(

033.32232

3.)1)(3(22

)1)(3)(3(21928

22

22222

2222

22

2233

=++=

++=++

+++++

+++

+++++

xxxxx

xxxxxxxx

xxxxxxxx

xxxxxx

xxxxxxxx

i chiu iu kin, kt lun bt phng trnh cho v nghim.Bi 32:Gii bt phng trnh : 21 2

2

log log (2 ) 0 ( )x x R > .

Hng dn:- iu kin: 2 22log (2 ) 0 2 1 1 1x x x > > < <

- Khi 22 2 21 1 1 1 1 1

log (2 ) 1 02 2 0

x x xx

xx x

< < < + 2 24 ( 2 4) ( 2 4) 3x x x x x x+ > + + (**)

+ TH 1: 1 5x + , chia hai vcho x > 0, ta c:

(**) 2 22 4 2 4

4 3x x x x

x x

+ + > +

- t2 2 4

, 0x x

t tx

+ = , ta c bpt: 2 4 3 0t t + < 1 3t < <

22

2

7 4 02 41 3

4 0

x xx x

x x x

1 17 7 65

2 2x

+ +< <

+ TH 2: 1 5 0x , 2 5 4 0x x+ < , (**) lun tha


15/21



Vy tp nghim bpt (*) l1 17 7 65

1 5;0 ;2 2

S + + =

Bi 34:Tm m bt phng trnh sau c nghim +

x 0; 1 3 :

( )2 2 2 1 2 0m x x x( x ) + + +

Hng dn:t 2t x 2x 2= + +do x [0;1 3] nn [ ]1;2t

- Bt phng trnh trthnh:

+

2t 2

mt 1

- Kho st hm st

g(t)t

=

+

22

1 vi [ ]1;2t

- Ta c:+ +

= >

+

2

2

2 20

1

t tg'(t)

(t ). Vy

tg(t)

t

=

+

22

1ng bin trn [ ]1 2;

2( ) (2) 3Maxg t g= =

- T:2

t 2m

t 1

+c nghim t [1,2]

[ ]tm g t g

1;2

2max ( ) (2)

3 = = . Kt lun:

2

3m

Bi 35:Gii bt phng trnh )(94117652 2 Rxxxxx ++++

++++

>

+++

++

+++

+ xxxxx

x

xxx .

Kt lun 1x

Bi 37:Gii bt phng trnh sau 2 5 3 2 4 1 5 6x x x x+ + > + + Hng dn:

2 5 4 1 3 2 5 6 0

1 1( 2 4)[ ] 0

2 5 4 1 3 2 5 6

2

+ + + >

+ + >+ + + +

+ t21

xt

x=

, khi bt phng trnh (1) trthnh: 21

3 2 02

tt t

t

>

+ Vi t < 1 th 22

1 1 (2)1

xx x

x< <

* 1 0 :x < bt phng trnh (2) ng

* 0 1:x< < bt phng trnh 2 22

(2) 1 02

x x x < < <

Tp nghim ca bt phng trnh (2) l 12

1;2

S

=

+ Vi t > 2 th 22

2 2 1 (3)1

xx x

x> >

* Bt phng trnh (3)2 2

0 2 5

54(1 )

xx

x x

> >

>

Tp nghim ca bt phng trnh (3) l 22 5

;15

S

=


17/21



Vy tp nghim ca bt phng trnh l 1 22 2 5

1; ;12 5

S S S

= =

Bi 39:Gii bt phng trnh: )(152)13( 23 Rxxxx ++>+

Hng dn:iu kin:3

1x

+ Bt phng trnh cho tng ng vi

)1(0)113)(213(

)213)(213()213)(13(

134)213)(13(

)13(2152)213)(13(2

2

++

++>++

+++>++

xxxx

xxxxxxx

xxxxx

xxxxxxx

+ Ta c3

1,0113 >+++ xxx nn

(1) )2(0)1()213(0

113

)1()213(>+>

+++

+ xxxx

xx

xxxx

Xt hai trng hp xy ra

+) Vi

>

0

10)1(

x

xxx th (2) 1

10

0

0134

0

0

2132


18/21



( )

22 2 2

2 2

2

2

x x 1 1 1x 1: x 5 x 5 x x 5 x

x 1 x 1 x 15 1

5 x 1 x 5 x 4x 5 x 5x 1x 5 x

5x

x 24

15x 40x 20 0

++ > + > + > + + >

> > + + > ++ +

>

> + >

Vy : x > 2

Bi 42:Gii bt phng trnh : ( )2 3. 2 1 2 1x x x x x +

Hng dn:K:1

x2

( )( )

( )

)

2

2

2 1 2 1 0

2 1 0 2 1 0

1 2;

+ + +

+ + +

+ +

BPT x x x x

x x vi x x

x

Bi 43:Gii bt phng trnh: + + < +0,2 0,2 0,2

log x log (x 1) log (x 2) .

Hng dn:iu kin: >x 0 (*).

+ + < +0,2 0,2 0,2

log x log (x 1) log (x 2) + < +20,2 0,2

log (x x) log (x 2)

+ > +2x x x 2 >x 2 (v x > 0).

Vy bt phng trnh c nghim >x 2 .

Bi 44:Gii bt phng trnh:2

20 4 2 4x x x x+ + + + Hng dn:iu kin: x 0 (*)+ x = 0 l nghim bpt (1)

+ x > 0 chia 2 vBPT cho x ta c:4 2

x 20 1 2 xx x

+ + + +

- t 22 4

t x x t 4xx

= + + =

Bt phng trnh thnh: 2t 16 2t 1+ 2 2

1t

t 32

t 16 4t 4t 1

+ +

Vi t 3 ta c:2

x 3 x 4;0 x 1x

+ <

Kt hp vi iu kin (*) v nghim x = 0 ta c tp nghim bpt l [ ]0; [ ; ]S 1 4= +

Bi 45:Gii bt phng trnh:2300 40 2 10 1 3 10

01 1 2

x x x x

x x

+ +


19/21


20/21


21/21


Bi 49: Gii bt phng trnh sau1 2

3 13

log (2 8) log (24 2 ) 0+ + + x x

Hng dn:

iu kin:x 1

x 2

2 8 0

24 2 0

+

+

>

>

(1)

( ) ( )x 1 x 23 3log 2 8 log 24 2+ +

x 1 x x

x 1 x 2 x x x

2 8 0 2 4 2 4

2 8 24 2 2.2 8 24 4.2 6.2 32

+

+ +

> > >

x2

16 164 2 2 x log

3 3

< <

Bi 50:Gii bt phng trnh 22( 3 3 2 ) 2 3 7 0+ + + x x x xHng dn:

iu kin :3

3 x2

( )

( )( )( )

( )

22 x 3 2 1 3 2x 2x 3x 5 0

1 3 2xx 3 42 x 1 2x 5 0

x 3 2 1 3 2x

2 4x 1 2x 5 0

x 3 2 1 3 2x(*)

+ + + +

+ + + +

+ + +

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