S = 0;1
4 u[4; + 0 3 )
pKXD: 1 - X > 0 <=> -1 < X < 1 ^ • V w , 1
1 3 x^ X
Bat phuong trinh <=> ^ -1 + 2 > , <=> ^ - 3 , _r + 2 > 0 . 1-x^ 7177 i - x 2 7 ^
Dat t =
t < l c=>
ta CO bat phuang trinh : t^ - 3t + 2 > 0 <=>
V-'
< l o V l - x ^ >x<=>
-1 < x < 0
0<x<l <^
t < l t > 2
l - x 2 > x 2
-i<x<o < * i' 3 ' rV
l o - l< x < - ^ .
t>2<=> • 2 o X > 2Vl7x^ ô • 0< x < l 2 x 2> 4 ( l- x 2 ) V 5
Vgy nghiem ciia bat phuong trinh da cho la: T = f I 1 ^ ' ' V 2 ; r 2 ]
Vi du 6: Cho phuong trinh %/x + Vl - x + Vx - x^ = m a) Tim m de phuang trinh c6 nghiem duy nhat j) Tim m de phuang trinh c6 nghiem.
Lot gmi
DKXD: 0 < X < 1 ' Gia su phuong trinh c6 nghiem duy nhat XQ tuc la ta c6 ' '
r — I I , r
v'^o + ~ '^o + V'^o ~ '^o) = t3 CO the viet lai la "
V ^ ~ ^ + sf^) + 7 ( 1 " = "1 do do 1 - X ( , cung la nghiem ciia phuong
Wnh da cho , ...
, (ằ>•.' -
'^o do phuong trinh c6 nghiem duy nhat thi xg = 1 - Xg <=> xg = - thay vao
CO m = V o i m J
I + 2V2 2
I + 2V2 r — / 2" 1 + 2V2
Vx + Vl - X + vx - X = ( )
^ ta CO phuong trinh Vx + Vl - x + Vx - x"^ = (*)
Phiin loai va pliumtg plidp s'"' •'<'>'<>_
, I J r~r-. c X + 1 - X 1 A p d u n g BDT cosi ta co Vx - x^ = ^ x ( l - x) < ^ = - Mat khac (V^ + V T ^ f = l + 2 7 x( l - x) < 2= > V ^ + 7 T^ < > / 2 Suy ra ^yx + ^/^ - x + V x- x ^ < , dang thuc xay ra <=> x = i Do do p h u o n g trinh (*) co nghiem duy nha't ^ . ^ ^ ^ ^ Vay m = — la gia tri can t i m . „ , j/ ' I ^j^,,^ ^ b) Dat t = N/^ + V T ^ ^ t^ = 1 + 2^ x( l - x )
Theo cau a ta co 1 < (Vx + ^l-xf = 1 + 2^ x( l - x ) < 2 S J S u y r a l < t < x/ 2 , j;-. j • j , , • , ^ Phuong trinh tro thanh t t ^ - l = m ằ t 2 + 2 t - l = 2m (**)
Phuong trinh da cho co nghiem khi va chi khi phuong trinh (**) co nghiem thoa man l < t< N/ 2
<=> D o thi ham so y = t^ + 2t -1 tren [ l ; V2 ] cat d u o n g thMng y = 2m . X e t h a m s o y = t 2 + 2 t - l tren \v,^'^ ^Mtmm^
i"'
t 1 ^
i"' y _ _ _ _ _ ^ l + 2V2
0 —
J, Suy ra p h u o n g trinh da cho co nghiem <=> 1 < 2 m < 1 + 2^
, 1 ^ ^1 + 2V2 hay <=> - < m <
' 2 2 i'l'"- •'!!•;
S " ' 3 : P h u o n g phap danh gia
D o i v o i p h u o n g trinh ta thuong lam n h u sau \
< Cach 1: T i m mot nghiem va chung m i n h no la nghiem d u y nhat.
Cach 2: Bien doi hSng dSng thuc dua ve bat p h u o n g trinh f (x) = 0 tron^
f (x) la tong cac b i n h phuong. ^ Cach 3: V o i p h u o n g trinh f(x) = g(x) co tap xac d j n h D
Neu f ( x ) > m ( x )
g ( x) < m ( x ) , Vx e D t h i f(x) = g(x) o f(x) = m(x) g(x) = m(x)
yj{ 7- Giai cac p h u o n g trinh sau
J j ^ + ^/jr^ = 6 b) +75774= ^ / 2 ^ + ^/3^
^ 3) , .
'^^^ • c Lcrigidi
3) DKXD: x < 2 . ,1,,,.;;,.,,; o . ' . „ , , r . .
Ta thay rang p h u o n g trinh co mot nghiem la x = - va ta chung m i n h do la nghiem duy n h a t . \l;f'"*' ^"v':'i V j ;i<)7 g r u w b g m n i l ônrn
Thgtvay ; ,|, • ; 7 r • ^ / f 7 ' -
* Voi X < - ta C O > 4 => / > 2 va
2 3 - x V 3 - X V 2 - X >
2- 3 , 2
= 4
• + > 6 =:> phuong trinh v6 nghiem.
• Voi ^ < X < 2 ta C O — ^ < 4 => . / — ^ < 2 va ằ =4 2 - 3 2 3 - x ^ ' - ^ v 3 - x - ^-"vri^ 2 Suy ^/jr~ ^/^r^ ^ ^ ^ phuong trinh v6 nghiem.
V|iy p h u o n g trinh co nghiem duy nha't x = - . , , 1 2 > / c,>""'0••;••(!•,- x|x
b ) D K X D : x > 0 x.*:'=xi;i Phuong trinh tuong d u o n g voi V2x + 3 - Vx + 4 = f x + 8 - 7 3 x
De thay X = 1 la nghiem ciia phuong trinh
Voi x > 1 ta C O 2x + 3 > X + 4 o V2x + 3 - N/X + 4 > 0
Va ( x- l ) ( 9 x + 8 ) > 0 o 9 x 2 - x - 8 > 0 o x + 8< 9 x 2 ô . f ^ - V 3 ^ < 0 Suy ra p h u o n g trinh v6 nghiem
Voi 0 < X < 1 ta CO 2x + 3 < X + 4 ô . V2x + 3 - V x + l < 0
Va ( x - l ) ( 9 x + 8 ) > 0 o 9 x 2 - x - 8 > 0 <::>x + 8> 9 x 2 c > f^ - 7 3^ > 0 Suy ra p h u o n g trinh v6 nghiem
Ygj^phuong trinh co nghiem duy nhat x = 1 . 8: Giai cac p h u o n g trinh sau
^ - 9 x + 2 8 ^ 4 V ^ b ) _ x^ 2 2 x+ V l± 2 ^ = 2- x 2 Laigidi
®KXD: x > l ' ' , . , .... • .
* nuong trinh tuong d u o n g voi
'ô^-10x + 25 + ( x - l ) - 4 V x ^ + 4 - 0 <:^(x-5)^+(7x71-2)^=0 (*) 349
V i (x - 5)^ + (N/X - 1 - 2)^ > 0 v o i moi x nen Phuang trinh (*) x- 5 = 0
<=> X = 5 Vay p h u o n g trinh c6 nghiem duy nha't x = 5 .
f l - 2 x > 0 1 1
b) D K X D : ^ ^ ^ ô - - < x < - M r .
Phuong trinh t u o n g d u o n g v o i ( > /T^ ^ + V l + 2 x ) = (2 - x^)
2
2 + 2 V l- 4 x ^ = 4 - 4 x ^ + x'* o ( V l - 4 x ^ - 1
x = 0
; <r>x = 0
V i- 4 x ^ - 1 = 0 . .' ; •' • -;'^r: • M
+ x'* = 0 y - (. / X -
Vay p h u o n g trinh c6 nghiem d u y nhat x = 0.
V i d u 9: Giai cac p h u o n g trinh sau 2x^ + X- 1
1 + 3VX + 1 b) \/x^ - 1 + X = Vx^^ - 2
a) Gia s u p h u o n g trinh c6 nghiem, k h i do nghiem cua no phai thoa man x + l > 0
x ( x - l ) > 0 C : > X 6 { - 1 } U [ 1 ; + M ) 2x^ + x - l > 0
R6 rang x = - I khong la nghiem cua p h u o n g trinh, ta xet x > I Phuong trinh da cho o 2x^ + x - 1 = Vx^ - x + 3^Jx{x^ -1) (*)
Ap d u n g BDT cosi ta c6: Vx^ - x < 3 7 x ( x ^ - 1 ) <
3(x + x 2- l )
Suy ra VP(*)
3(x + x ^ - l ] ,
• * ) < - - + - ^ ^ = 2 x 2 + x - l = VT(*) 1 ± V 5 Dang thuc xay ra k h i va chi k h i x - x - 1 = 0 ằ x =
T h u lai p h u o n g trinh ta tha'y x = llJl. la nghiem cua p h u o n g trinh Vay p h u o n g trinh c6 nghiem duy nha't x = 1 + 75
350
D K X D : x 3 - 2 > 0 < = > x > ^ Gia s u p h u o n g trinh c6 nghiem
Su d u n g bat dMng thuc cosi, ta duoc ^x^ - 1 < 2 ( x - l ) + (x + l ) + 4 ^ x + l_
6 2 pCet hop v o i p h u o n g trinh suy ra + x ^ Vx"' - 2
^ 4( x ^ - 2) < ( 3 x + l)2 ô ( x- 3 ) ( 4 x 2 + 3 x + 3) < 0 c ^ x< 3 ! • : N h u vay ta c6 ^ < x < 3. (**)
Ta CO ^^x^ - 1 > X - 1 o X + 1 > (x - 1 ) 2 ô . x(3 - x) > 0 (dung v o i d k (**)) va Vx^ - 2 < 2x - 1 c=> (x -3){x^ - x + 1) < 0 (dung v o i d k (**))
Suy ra ^x^ - 1 + x > 2x - 1 > Vx^ - 2 . . .
Dang thuc xay ra k h i x = 3. T h u lai ta tha'y x = 3 la nghiem ciia p h u o n g trinh da cho
Vay p h u o n g trinh c6 nghiem duy nhat x = 3.
Nhan xet: V o i dieu ki^n xac djnh cua p h u o n g trinh t h i viec danh gia cua chung ta kho khan, d o i k h i la khong the danh gia v i mien ciia bien luc do rgng khong d a m bao cho viec danh gia. D o do rang buoc them dieu kien doi v o i nghiem cua p h u o n g trinh giup chiing ta thuan Igi trong danh gia t u do giai quye't duoc bai toan.
• ZBAlTAPLUYeNTAP _ ^ x W d n h i ^ n o . K l q r Bai 4.110: Giai cac bpt sau :
a) V x^ < 2 x - 1 c) V 3 x - 2 > 4x - 3
0 > S . - ^ C
' i . , . x , < 0 ^ b) N/X^ - x + 1 < x + 3 - " ^ ^ f '
a) Bpt
^) Bpt
2 x - l > 0
x - 3 > 0 <=>
x - 3 < ( 2 x - l ) 2
x^ - X +1 > 0 y x + 3 > 0
x 2- x + l < ( x + 3)^
d) V3x2+x-4>x + l
Hu&ng dan giai £y c> € 77% o (")
x > i ,,,an i\ *
x>3 o x> 3 ' 4x - 5 x + 4 > 0
• ằ x > — 7
351
c) Bpt<=> 4 x - 3 < 0 4 x - 3 > 0
3 x - 2 > 0 3 x - 2 > ( 4 x - 3 ) ' <=>
2 3
— ^ X < —
3 , 3 - < x < l , . .
4 )i.,;r; •(
d) Bpt<:>
x + l < 0 3 x ^ + x - 4 > 0
X + 1 >0 A ô u > { 3x^ + x - 4 > ( x + l)^
3
x> 1 + 74T.
Bai 4.111: Giai cac bat phuong trinh sau. 1 ^ / ' - f
a) (x^ - 3X)N/2X^ - 3 x - 2 > 0 - b) x^ + 3x +1 < (x + 3 ) N / X ^ Huang dan giai ' . ; a) Ta xet hai truong hop
T H 1: 2x^ - 3x - 2 = 0 ô x - 2,x = . Khi do BPT luon dung
fi' T H 2: B p t ô 2 x 2 - 3 - 2 > 0 x^ - 3 x > 0
x < - - V x > 2 1 ^ - 2 <=>x<— V x > 3 . x < 0 V x > 3 ^
' 1 - o c ;
V 2 u{2}u[3;+oo).
Vay nghi^m ciia Bpt da cho la: T =
b) Bat phuong trinh o x(x + 3) - (x + 3) 7 x 2+ l + l < 0 o ( x + 3 ) ( x - Vx^ +1) + (N/X^ + 1 )2 - X ^ <0
< ^ ( N / X2+ 1 - X J | N / X2+ 1- 3J < 0 (*)
Do Vx^ +1 - x > V x ^ - X = |x| - X > 0
= > ( * ) ằ V x^ + 1 <3<x>x2 < 8 o- 2 7 2 < X< 2N/ 2 .
V|y - 2 V 2 < x < 272 la nghi^m ciia bat phuong trinh da cho. ^ . Bai 4.112: Giai cac bpt sau : ^
a) V 2 x 2- 6 x + l - x + 2 > 0 c) Vx + 2 - Vx + 1 < N/X
b) Vx + 3 > V 2 x- 8 + N/7-X d) 2x^
( 3 - V 9 + 2x) Huang dan giai
:x + 21
x- 2 < 0
a) B p t< ^ V 2 x^ - 6 x . l > x- 2 o | ^ ^ , _ ^ ^ ^ ^ ^ ^
352
Cfi/ TNHHMTV DVVU K i u i H ^ Vift
hoac
' x - 2 > 0
• ( 2 x 2 - 6 x + l ) > ( x - 2 ) 2 ^ ' x < 2
x <
x > 3 + V7
hoac x > 2
x ^ - 2 x - 3 > 0 x <
x > 3 3 - V 7
b) DKXD:
x + 3 > 0 "
2 x - 8 > 0 < = > 4 < x < 7 7 - x > 0
Bpt ô x + 3> ( V 2 x- 8 + V 7 ^ ) o 3 > - l + 2 7 ( 2 xx2 - 8 ) ( 7 - x ) o 2 > 7 ( 2 x- 8 ) ( 7 - x ) o 4> -2x2 + 2 2 x- 5 6
X <5
X > 6
•4<x<5 ^ : o x ^ - l l x + 3 0 > 0 o
Doi chieu dieu kien ta nghi^m bpt la
6 < x < 7 c) D K X D :
X + 2 > 0 ' T • x + l> 0 <x>x>0
x> 0 ,„ ,
^pt O V X + 2<N/X + 1 + N ^ C J . X + 2 < 2X + 1 + 27(x + l ) x
. r i- x< o
< = > l - x < 2 J ( x + l ) x ô•< „ - - .
3 + 2^ ! f - < • j l - x > 0
hoac -! , o ( 1 - x ) <4x(x + l )
X < — - 3 + 2V3
•<x 1. / . - V'
^ 1 chieu dieu ki?n ta nghi^m bpt la x > ^ ' ^ ^ ^
^) D K X D : f9 + 2 x > 0
3 - 7 9 + 2 x ^ 0 2
X;tO
Bp 2x2(3 + ^/9T2^)
t o i < X + 21
4x2
Vv Jq<i !
oV9 + 2 x < 4 o x < - 2
9 7
— < X < —
2 2
X ^ 0 Doi chieu dieu i<ien ta nghiem bpt la
X ^ 0
- ^' ' v . ,
Bai 4.143: Giai cac bat phuong trinh sau: H ' \
b) \lx^ - 3x + 2 +\/x^ - 4 x + 3 >2\/x^ - 5x + 4^
Huang dan giai , V-3x^ + X + 4 + 2 -
a) — < ^
a) D K X D :
4 ! - 1 < x < -
x ^ O
Voi 0 < x < - : BPT ô 3
V_ 3 x 2^ x - f 4 + 2 ^ ^ ^ ^ _ 3^ 2^ ^ ^ 4 ^ 2 x - 2
"FfJ
2 x - 2 > 0 X> 1
_3x2+x + 4 < ( 2 x - 2 f [ 7 x 2 - 9 x> 0 9 o X > - 9 4
Suy ra nghiem cua ba't phuong trinh la -<x<- Voi - 1 < X < 0 : bpt luon diing
Doi chieu dieu ki?n ta c6 nghiem bpt la f x 2 - 3 x + 2 > 0
- l< x< 0 9 ^ 4
- < X < -
7 3
b) DKXD: x^ - 4 x + 3>0<:=>
x2 - 5 x + 4> 0
x> 4
x < l v S
0 < I
B p t o 7( x- l) ( x - 2 ) + V ( x- l) ( x- 3) > 2 7 ( x- l) ( x- 4 ) De tha'y x = 1 la nghiem ciia bpt.
+ Voi x < 1: Bpt c> 7(1 - x)(2 - x) + ^(1 - x)(3 - x) > 2^(1 - x)(4 - x) o N/ 2 -X + V 3- X > 2 V 4- X
Ta CO : >y2-x + V 3 - x < N / 4 -X + N/ 4 -X = 2 V 4- x Suy ra X < 1 bpt v6 nghiem .
+) Voi X > 4 : bpt o V x- 2 + N / X- 3 > 2>yx-4
T a c o : >yx - 2 + Vx - 3 > Vx - 4 + Vx - 4 = 2Vx - 4, Vx,x > 4 Suy ra : x > 4 bat pt luon dung . )
"x = l x > 4 Vay nghif m cua bpt la :
4.114: Giai cac bat phuong trinh sau:
g)V3x^+6x + 4< 2 - 2 x- x 2 b) 2x2 + 4x + sVs - 2x - x^
^) Vix^ + 5x + 7 - V 3 x 2+ 5 x + 2 > 1 d) Vx + 2 > / ^ + V x - 2 V x ^ > -
e) 5 V ^ . - f < 2 x . J - . 4 0 ^ - 2 . / ^ > 3 g ) x . ^ > 3 ^
2Vx 2x x + 1 V X 12
a) Dat
Huang dan giai rin/riJor'n at: t = V3x2 + 6x + 4 , t > 0= > x 2 +2x = ^—^ . K
Bat phuong trinh tro thanh
t < 2 ~ ô t 2 + 3 t - 1 0 < 0 o 0 < t< 2( t > 0 ) , . * ,
Ta CO V 3 x 2 +6x + 4 < 2 <ằ 3x^ + 6x + 4 < 4 o 3x2 + 6x < 0 o -2 < x < 0
Vay nghiem bpt la -2 < x < 0 .
b) DKXD: -3 < X < 1 ' ^ ' '
Dat: t = V 3 - 2 x- x 2, t > 0^ t 2= 3 - 2 x- x 2^ 2 x + x 2 = 3- t 2 <x Bat phuong trinh tro thanh i
2 ( 3 - t 2 ) + 3 t> l c^2t2 - 3 t- 5 < 0 c : > 0 < t < - ( d o t > 0 ) Ta CO V 3 - 2 x- x 2 < - ô .
2
-3 < X < 1
„ - 2 25 ô ' - 3 < x < l 3 - 2 x - x < —
xV
Vay nghiem bpt la -3 < x < 1.