. Tru hai phuong trinh ciia h^ ta dugc: ,,
BAT PHL/ONG TRINH VO TY
Giai bat p h u o n g trinh v6 ty luon dup-c xay d u n g dya tren nen tang la giai phuong trinh vot tỵ
Vi vay de giai dugc cac bat phuong trinh v6 ty thi truoc het can nam chSc phuong phap giai phuong trinh v6 tỵ
Ngoai ra can chii y nhiing diem sau: + Luon dat dieu ki$n truoc khi giai
+ De binh p h u o n g 2 ve cua bat phuong trinh ma khong lam doi dáu bat phuong trinh thi dieu ki^n la 2 véphai khong am
+ Khi bat phuong trinh c6 an 6 mau só ta can de y xem c6 the danh gia mau so (+) hay (-) dya vao dieu ki#n xac djnh hay khong? Qua do de lam don gian bat phuong trinh.
Vỉc danh gia mau soco the dya vao dáu ciia tam thuc bac 2, d u n g bat dSng thuc, ham sọ..
+ K h i giai cac bat p h u o n g trinh v6 ty co ban can l u u y:
<=> g ( x ) > 0 f ( x ) > 0 f ( x ) < g 2 ( x ) V f W > g ( x ) ' f(x) > 0 g ( x ) < 0 g ( x ) > 0 f ( x ) > g 2 ( x )
V i d y 1: Giai cac bat p h u o n g trinh sau:
c) V x ^ - 3 x + 2 + V x ^ - 4 x + 3> 2 7 x 2- 5 x + 4 • < d) \/x^+x- 2 + 7 x ^ + 2 x- 3 < V x ^ + 4 x - 5 e) V2X2+12X + 6 - N/ 2X- 1 >X + 2 0 7 x ^ - 2 x ^+ X < X N / X + V x ^- 2 x GiAi: 3)- Dieu ki^n: x^ - 3 x + 2 > 0 x^ - 4x + 3 > 0 o x^ -5^+4>0 x > 4 x < l
Truong hgp 1: x > 4 . Ta viét Igi bat phuong trinh thanh: / ( x - l ) ( x- 2 ) + V( x - l ) ( x - 3 ) > 2 7( x - l ) ( x - 4 ) . . .„ Do - 1 > 0 nen bat phuong trinh tuong d u o n g voi:
Vx- 2 + V x - 3 >2N / X - 4 <=> Vx- 2 - V x - 4 > V x - 4 - V x - 3 .
TAi lifu 6n thi d^i hgc sang t^o va giai PT, hat PT, hf PT, bat DT - !\'<^u,i,-„ i,imgKieh_
Bat phuong trinh nay luon dung do x - 2 > x - 3 > x - 4 . Truong hpp 2: x < 1 . Ta viet lai bat phuong trinh thanh:
7(1 - x)(2 - X ) + V(l - x)(3 - X ) > 27(1 - x)(4 - X ) .
Voi X = 1 bat phuong trinh nghi^m diing. Voi X < 1 . Bat phuong trinh tro thanh:
V 2- x + V 3- x > 2 V 4- x o 7 2- x - V 4 - x ^ V 4 - x- 7 3- x
Vi 4 - x > 3 - x> 2- x = > V T< 0, V P> 0 nen bat phuong trinh v6 nghi^m. |'x = l
Ket lu^n: Nghi^m ciia bát phuong trinh la: x^4
b)
x^ + x - 2 > 0
x^ + 2 x - 3 > 0 o x > l x < - 5 x^ + 4 x - 5 > 0
Truong hgp 1: x < - 5 . Ta viet l^i bát phuong trinh thanh:
7(1 _ x)(-x - 2) + 7(1 - x)(-3 - x) > 7(1 - x)(-5 - x) o 7- x- 2 + 7- X- 3 < 7- x - 5
Do - X - 2 > - X - 3 > - X - 5 nen bat phuong trinh v6 nghi^m
Truong h^p 2: x > 1 .
Ta thay x = 1 thoa man bat phuong trinh:
Xet X > 1 . Ta viet lai bat phuong trinh thanh:
7( x - l ) ( x + 2) + 7( x - l ) ( x + 3) < 7( x - l ) ( x + 5) o 7 x + 2 + 7 x + 3 < 7 x T 5 Binh phuong 2 ve bat phuong trinh ta c6: x + 27(x + 2)(x + 3) ^ 0 , b. phuong trinh nay v6 nghi^m do VT > 0
Ket luan: x = 1 c) Dieu ki^n: x ^ ^
Cach 1: Ta viet l?i bat phuong trinh thanh:
72(x + if + 2(2x -1) > X + 2 + V2x - 1 . Dat a = 7 2 x- l , b = x + 2=:>a,b>0.
Bat phuong trinh c6 dang: 72(â + b ^ ) > a + b o ( a - b ) ^ > 0 < = > a 7 t b .
Xet a = b<=> V 2 x - 1 ^ x + 2<»x^ - 6 x + 5 = 0 o x = l x = 5
Vayd 1 \|i;5)
Vay de a ;t b ta can CO X e —;+oo \{1;5)=:>S = L /
Cach 2: Ta viet lai bat phuong trinh thanh: 72x^ +12x + 6 > \l2\- \ x + 2 . Do hai ve bat phuong trinh deu khong am. Binh phuong 2 véta thu du^c:
<:> 2x^ + 12x + 6 > 2x - 1 + 2(x + 2)N/2X-1 + X^ + 4X + 4 <^ + 6x + 3 - 2(x + 2)N/2X -1 ' i ^ + 6x + 3-27(x^ +4x + 4) ( 2 x-l) > 0 . . • ,' X Gia su x^ + 6x + 3 = m(x^ + 4x + 4) + n(2x -1): r m = i [It ) n = l Bat phuong trinh tro thanh:
(x^ + 4x + 4) + (2x -1) - 27(2x - r){x^ + 4x + 4) > 0 •ii I E , . , , I" •ii I E , . , , I" V2X-1 -7(x2 +4X + 4) '1 >0 ... £ o 72x - 1 - Vx^ + 4x + 4 / 0 Ket luan: S = d) Dieu kifn: x ^ l x*5 1 -;.=c \{i;5) x 2- 2 x > 0 x > 0 D|t a = 7 x 2- 2 x {• b= 7 ^ o x> 2 a > 0 b > 0
Khi do bat phuong trinh tro thanh: ab + b^ ^ b'' + a <=> (b - l)(a - b^) < 0 b > l x > l
— <=>x>l 2x < x
Truong h g p l: i , o J r——
[ a- b 2 < 0 [ 7 x 2 - Ket hg-p dieu ki^n ta suy ra x > 2
Truorng hop 2: <^ , o <^ r - o V N [ a - b ^ ^ O [ 7 x 2- 2 x > x
Ket luan: S = [2;+oo) ••(«-,(Si «»•-
2: Giai cac bat phuong trinh sau:
3) x2 >(i_ 7;^)(2x- 3 7 ^ + 3) b) x + l + 7 x 2 - 4 x + l> 3 7 ^
Tai lien on thi tUii hoc sdiig tao va giai PT, hat PT, hJT'T,T)al t)l- \ ' . ; i i r / r ; i Vniitg Kien
G i i i :