I hl t jl tj <
a) Dieu kifn :x < 1 Ta viétlai bat phuong trinh thanh.
1 , 3x ^ X
- - 1> - F 2 o
3x
1-x^ - + 2 > 0 .
D^t t = , ^ > 0. Ta CO bat phuong trinh moi t^ - 3t + 2 > 0 <=>
V l - x ^
Truong hpp 1: t < 1 <=> , ^ < 1 <=> V l - x ^ > x
V l - x ^
+ Neu - 1 < X < 0 bat phuong trinh nghi^m diing.
+ Neu X > 0 binh phuong 2 ve ta thu duc^c: 1 - x^ > x^ <=> x <
t < l
t>2
Suy ra 0 < X <
Tom lai trong truong hcp-p nay ta c6: Sj =
Truong hgip 2: t > 2 > 2 o <=> x>—— x 2 > 4 - 4 x 2 5 x > 0 pg'y rang: 2^x2 - x + l j = X — 1 V 2 , + -^>1=>1-J2fx2-x + l ) < 0 '
Bát phuong trinh tro thanh: x - Vx < 1 - ^l{x^ - x + 1)
Den day ta c6 cac huong xu ly nhu sau:
» Cachl:
+ Xet x = 0. Bat phuong trinh khong thoa man.
+ Xet x > 0. Chia hai ve bat phuong trinh cho Vx ta c6:
' 1^ x + —
X - 2 <0.
I Dat t = Vx - =:> t^ = X + i - 2 thi bát phuong trinh tro thanh: '"'^ ' ' V X X .v;^= - l + Vs 3-V5 o X = • .v;^= - l + Vs 3-V5 o X = •
2 2
Cach 2:Bát phuong trinh dug^c viét lai nhu sau:
>/2(x2-x + l) < V x + l - x .
D5t 4^ = a , 1 - X = b thi bat phuong trinh c6 dang:
N & + 2 b 2 <a + b « < •J a + b>0 r- J - ^\ ^ . <=>a = b > O o v x = l - x o Vx = ( a - b r < 0 2 «>x = 3-V5
Cach 3: Dat Vx = t > 0 . Bat phuong trinh tro thanh:
I [ - t 2+ t + l > 0
V2t^- 2t^ + 2 <-t'' + t +1 <=>
2t'*-2t2+2^(-t2+t + l ) 2
- t ^ + t + l > 0 0
Tai U(u on thi dai UQC sang taa va gidi PT, batPT,lifPT, hoT Dl \['^uur>i TniiigKien
-t^+t + l> 0 - i + Vs 3 - S
t 2 + t - l = 0 2 2
c) D i e u k i # n : x > 0 . De y rang v o i x > 0 thi ^2(x^ +6x + l ) - l > 0 Bat p h u o n g trinh tro thanh: x - 1 + yJ2{x^ +6x +1) - 4Vx > 0 Ta tháy: x = 0 la nghi^m ciia bat p h u o n g trinh: ..,^,.^ỵ.. ^
Q u a 2 ve cho Vx > 0 ta t h u dxxgc: - 4 + + 1 6 > 0 .
Dat t = x - 1
Ta thu duq/c bat p h u a n g trinh: V2t2+16 > 4 - t
+ Neu t > 4 bat p h u a n g trinh luon thoa man. + Neu t < 4 bat p h u o n g trinh tuang d u o n g v o l :
2 t ^ + 1 6 > t 2 - 8 t + 1 6 c ^ t 2 + 8 t > 0 » t > 0 t < - 8 Ket hg-p 2 t r u o n g hp-p ta suy ra t > 0 t < - 8 + Neu t< - 8 < » ^ / x- 4- < - 8 « x + 8 ^ / x - l < 0 o ^ / x < ^ + ^yi7<»x<33-8^/l7 ket h(?p dieu kỉn x^ 0 = > 0 < x < 3 3 - 8 ^ J V 7 + Neu t > 0 <=> X > 1
Ket luan: S = [O; 33 - 8N/37 ) u (1; +oo)
[ x > l .
1 3 - 3N/ ^ ^ 0 '
Ta viet lai bat phuong trinh thanh:
d) Dieu ki^n:
x ^ - 2 x 2 - 4 0 x ^ - 2 x 2 - 4 0 - 1 3 x + 3 x V ^ ^ _ .
- < X O ; = =1 3 - 3N / ^ < 0 .
x 2 - 2 x - — - 1 3 - 3 N / X ^
Chia hai vébat p h u o n g trinh cho x ta thu dupe: — 7 = ^0
13-3x/)n Xet ham so f(x) = x^ - 2x - — - 1 3 + 3 V x ^ tren f l ; +00) Xet ham so f(x) = x^ - 2x - — - 1 3 + 3 V x ^ tren f l ; +00)
^c6f(x) = 2 x - 2 + i J + - i = > 0 . X -
p o do ham so f(x) dong bien tren l;-i-oo) irý ifsi i
Mat khac ta c6 f(5) = 0 suy ra f(x) luon ciing dau ho|ic cung trỉt tieu v o i
Bat p h u o n g trinh t u o n g d u o n g v a i : = = = - ^ 0 . 1 3 - 3 V X - 1
Truong hg-p 1 : x > 5 . Bat phuang trinh t u o n g d u o n g v a i 1 3 - 3 > / x - l < 0 178
13 < 3 V x - l •» X > suy ra Sj = 178
I 9 -;+oo
Truong hgp 2: x < 5 . Bat p h u o n g trinh tuong d u o n g v o i 1 3 - 3 V ^ > 0
. 278 < : : > 1 3> 3 V x- l o x < suy ra = [ l ; 5 ) < : : > 1 3> 3 V x- l o x < suy ra = [ l ; 5 )
ri78 1
l;5)u ;+oo I 9 ' J I 9 ' J
Ket hg-p lai ta c6 nghi^m a i a bat phuang trinh la: S = 1;5) u N h ? n x e t :
Trong l o i giai tren ta da six d y n g cac t i n h chat quen thuQc:
f(x ) - f ( x )
+ Neu f(x) dong bien tren D thi — — > 0 v o i m g i x, x, € D
X 1 - X 2
ffx ) —ffx )
+ Neil f(x) nghjch bien tren D thi — < 0 v o i m p i X j ^ X j € D
X j - X j
Dieu do ciing c6 nghla la:
+ Neu f(x) dong bien tren D m a f(a) = 0 t h i f(x) l u o n cixng dau hoac ciing tri^t tieu v o i x - a
+ Neu f(x) nghjc bien tren D m a f(a) = 0 t h i f(x) l u o n ciing dau hôc ciing trif t tieu v o i a - x
V i 5: G i a i cac bat p h u o t i g t r i n h sau: S - x- V s- x ^ a) ^ 2 x - 7 x - 5 cos cos ^0 b) \/2x2+llx + 15 + V x 2 + 2 x - 3 > x + 6 c) V ^ ^ , 9 x W 3 2 _ ^- - " ^ ' 16 d ) 2( v r^ + N/ r^ ) + V i - x 2 <i--x^+5.
Gi4i:
a) V6i nhung bát phuong trinh c6 an 6 mau só vi#c quan trpng dau tien dat rg la: L i ? u ta c6 the danh gia dau ciia mau só de tach mau só ra khoi baj