tOM T A T L Y T H U Y E T . 0inh nghia bat phuong trinh mot an
Cho hai ham so y = f (x) va y = g(x) c6 tap xac djnh Ian lup-t la Df va Dg . pat D = Df n Dg . Menh de chua bien c6 mot trong cac dang f (x) < g(x),
f(x)>g(x), f ( x ) < g ( x ) , f(x)>g(x) dugc goi Va bat phuomg trinh mot an;
X du<7C g9i la an sd'(hay an) va D goi la tap xac dinh cua bat phuong trinh.
XflSD goi la mQt nghiem cua bat phuong trinh f(x)<g(x) neu f(xo)<g(>^o) 'a menh dediing.
Giai mot bai phuong trinh la tim tat ca cac nghiem (hay tim tap nghiem) ciia bat phuong trinh do.
Chu y: Trong thuc hanh, ta khong can viet ro tap xac dinh D cua bat phuong trinh ma chi can neu dieu ki?n de x € D . Dieu ki^n do goi la dieu ki?n xac djnh cua bat phuong trinh, goi tat la dieu kien cua bai phuong trinh.
2. $dt phuong trinh tuong duong, bien doi tuong duong cdc bat phuong trinh.
a) Djnh nghia: Hai bat phuong trinh (eiing an) dugc goi la tuong duong neu chung CO cung tap nghiem.
Ki hieu: Neu f j ( x ) < g i ( x ) tuong duong voi f2(x)<g2('') thi ta viet flW<gi(x )<^f2 (x)<g2(x)
' Phep bien doi khong lam thay doi t^p nghifm cua bat phuong trinh gpi la
phep bien doi tuong duong. --— - — r
D}nh ly va h | qua: ^
0}nh ly 1: Cho bat phuong trinh f(x)<g(x) c6 tap xac djnh D ; y = h(x) la ham so xac dinh tren D . Khi do tren D, bat phuong trinh da cho tuong duong voi bat phuong trinh sau , , ,
1) f(x) + h ( x ) < g ( x ) + h(x) ^ . 2) f(x).h(x)<g(x).h(x) neu h(x)>0 voimQi x e D
^)f(x).h(x)>g(x).h(x) neu h(x)<0 voi mpi x e D
qua: Cho bat phuong trinh f (x) < g(x) c6 tap xac dinh D . Khi do
^ ) f ( x ) < g ( x ) ô f 3 ( x ) < g 3 ( x ) , r::.r-~-r:.'r~rr ^
^) f(x) < g ( x ) ằ f 2 (x) < g2 (x) voi f (x) > 0, g(x) > 0, Vx e D y: Khi giai phuong trinh ta can chii y . .j.„..,,.-....-,a,„
• Dat dieu kien xac dinh(dkxd) cua phuong trinh va k h i t i m duoc nghj^
cvia p h u o n g trinh phai doi chie'u voi dieu kien xac djnh.
• Doi voi viec giai bat phuong trinh ta thuong thuc hien phep bien doi tu(>, duong nen can luu y toi dieu kien de thuc hien phep bien doi tuong duorig
B. CAC DANG TOAN VA PHl/gNG PHAP G I A I .
II D A N G T O A N 1 : T / M DIEU KIEN XAC DINH CUA BAT PHUONG TRi^
<Phuang phdp gidi.
Dieu kien xac djnh cua bat phuong trinh bao g o m cac dieu kien de gia ciia f ( x ) , g ( x ) C L i n g duoc xac djnh va cac dieu ki^n khac (neu c6 yeu . t r o n g d e b a i ) ,;
Dieu kien debieu thuc . JUV) xac djnh la f ( x ) > 0
1 f ( x )
1
xac dinh la f ( x ) ^ 0 xac d j n h la f ( x ) > 0
e a r c A c vf DU MINH HOA
V i d u 1 : T i m dieu kien xac djnh ciia phuong trinh sau:
^ ^ b) V 4 - 2 x >
a) x +
4 x ^ - 9 < 1 x + 1
x ^ - 2 x - l Lai gidi
a) Dieu kien xac dinh cua bat phuong trinh la
, 7 9 3
^ „ 4 x - 9 ^ 0 o x ^ ^ - c ^ x ^ t -
b) Dieu kien xac dinh cua bat phuong trinh la 4 - 2 x> 0 x < 2
x ^ - 2 x - l ; i 0 ^ |x;^l±V2
x < 2 x ^ l - V 2
V i d u 2: T i m dieu kien xac dinh cua bat phuong trinh sau r o i suy ra nghiem cua no: - .• - a) 2x + 7 x - 3 > 2 7 3 - X + 3
c) ^f^- 1 1
N / X- 2 > y x - 2 + 2 ,0
b) + 4 x - 4 <27-3x^
d) ^(x-l)^(3-4x)-5x>74?
pieu kien xac djnh ciia bat phuong tri Lai gidi
, [ x - 3 > 0 r x > 3
inh la < O r^'^ i ^<=>x=3 3 - x > 0 x < 3 Thu vao bat p h u o n g trinh thay x = 3 thoa man
ilU^ly tap nghiep ciia bat phuong trinh la S = |2| - ^ - r , , t . Dieu kien xac djnh ciia bat phuong trinh la ' ' '
^x^ + 4 x ~ 4 > 0 < = > - ( x - 2 ) ^ >Oci> X = 2
Thay x = 2 vao thay thoa man bat phuong trinh Vay tap nghiep cua bat phuong trinh la S = 13}
x > 0
A Dieu kien xac djnh ciia bat phuong trinh l a i ^ " ^ < = > < | ' ^ ^ ^ o x > 2 [ x - 2 > 0 [ x > 2
Vol dieu kien d o bat phuong trinh tuong d u o n g voi A/X < 2 <=> x < 4 Doi chieu v o i dieu kien ta thay ba't phuong trinh v6 nghiem.
Vay tap nghiem ciia bat phuong trinh la S = 0 d) Dieu kien xac djnh ciia bat phuong trinh la
De thay x = 1 thoa man dieu kien (*).
f 3 - 4 x > 0
( x - l) 2( 3 - 4 x ) > 0
lo 4 x - 3 > 0 Neu x ^ 1 t h i ( * ) ô
4 x - 3 > 0
3
X . 3 4
4
' - - r
.-c Vay dieu kien xac djnh ciia bat phuong trinh la x = 1 hoac x = - 4 Thay x = 1 hoac x = - vao bat phuong trinh thay deu thoa man.
Vay tap nghiem ciia bat p h u o n g trinh la S = 2. BAl T A P L U Y $ N T A P
4.55: T i m dieu ki^n xac djnh ciia bat phuong trinh sau roi suy ra tap f^ghiem ciia no:
^) + V T ^ < vr:^ + 2
S x < l
b) 7( x - l) 2( 2 - x ) ( x - 2) > - 7 Huang dan gidi
b) x - l , x = 2 ^ ^ • *
DANG TOAN 2: XAC DINH CAC BAT PHUONG TRINH TUONG BUmo]
VA GIAI BAT PHUONG TRINH BANG PHEP BIEN DOI TUONG. \
^huxmg phdp giai.
De giai bat phuang trinh ta thuc hi^n cac phep bien doi de dua ve phuang trinh tuang duang voi phuang trinh da cho dan gian han tror, vi^c giai no. Mot so' phep bien doi thuong su dung
• Cong (tru) ca hai vecua bat phuong trinh ma khong lam thay doi dieu kien
xac djnh cua bat phuong trinh ta thu du^c bat phuong trinh tuang duong bat phuang trinh da cho.
• Nhan (chia) vao hai ve cua bat phuang trinh voi mgt bieu thuc duangihoac ludn dm) va khong lam thay doi dieu ki^n xac dinh ciia phuorij^
trinh ta thu duoc bat phuang trinh cung chieu (hoac ngucec chieu) tuong duang voi bat phuong trinh da cho.
• Binh phuang hai vecua bat phuang trinh (hai ve luon duong) ta thu duoc bat phuang trinh tuang duang voi bat phuang trinh da cho.
• Lap phuong hai ve cua bat phuong trinh ta thu duac bat phuong trinh
tuong duang vai bat phuang trinh da cho. Q.
Eai.CACVlDUMINHHOA
V i dy 1: Trong cac bat phuang trinh sau day, bat phuang trinh nao tuong duang voi bat phuang trinh 3x +1 > 0 (*):
1 1 X X
a) 3x +1 > b) 3x +1 + , > , x - 3 x - 3 V3xLai giai + 1 N/3X + 1
' Taco 3x + l > 0 < = > x> - i , •
3
a) 3x +1 — > — (1) khong tuong duong 3x +1 > 0 vi x = 3 la nghi?"^
x - 3 x - 3
cua bat phuang trinh (*) nhung khong la nghi^m ciia bat phuang trinh (1) b) 3x + l + - 7 = ^ = > - p ^ = < = > 3 x + l > 0 < = > x> - i
V 3 x + 1 V3x + 1 3
'^^ Do do 3x +1 + , ^ > , ^ tuang duong 3x +1 > 0.
^ / 3 ^ V 3 ^ ^
V i dy 3: Khong giai bat phuang trinh, hay giai thich vi sao cac bat phuang trinh sau nghi^m diing vai moi x.
a) 7|x-l| + x2 > 2 x - l b)-2^^ (x + l ) ^ < ^ x^+1 ^ ' x ^ + l
Lot giai
BPTc ^VF-^ + x^ - 2 x + l >0o7 |x-l|+(x-l )2 > 0
Do ^\x-l\>0, ( x - l) 2> 0 vai moi x nen 7|x-1| + (x-1)^ >o voi moi x. I
•^y bat phuong trinh nghi^m diing voi mpi x.
p T o - ( x + l) 2 <0c:>(x + l) 2 >0 (diingvoimoi X ) V$y bat phuang trinh nghiem diing voi mpi x.
p 2. B A I T A P L U Y $ N T A P
pii 4.56: Trong cac bat phuang trinh sau day, bat phuang trinh nao tuong duong vai bat phuang trinh 3x +1 > 0:
1 1
3X + 1 + >
x + 3 x+3 Taco 3x + l > 0 < = > x > - -
3
1 1
a) Taco 3x + l + >
b) 3x +1 + Vx +1 > Vx +1 Huang dan giai
f X;^-3 'xvt-3 x+3 X+ 3 3 x+ l > 0
1 <=>x> — 1
x > — 3 3
1 1
Do do 3x +1 + > tuong duong 3x +1 > 0 x + 3 x + 3
b) 3x +1 + Vx + 1 > Vx + 1 <=>I ^'^^^^ ^
3x + l > 0
r x > - i
1 c > x >
x > — 3 3
Do do 3x +1 + Vx + 1 > Vx + 1 tuong duong 3x +1 > 0
8a> 4.57: Khong giai bat phuong trinh, hay giai thich vi sao cac bat phuang trinh sau v6 nghiem.
N ^ > v - x - 4 b) NArrT <-x2+x-i
Huong dan giai x + l > 0
. o • x > - l X <-4 - x - 4 > 0
Suy ra bat phuang trinh v6 nghiem.
"Ta CO Vx + 1 >0, - x ^ + x - 1 ^
khong ton tai gia trj nao aia x
• X
-<0 ''Uy ra ba't phuang trinh v6 nghiem.
4.58: Khong giai bat phuong trinh, hay giai thich vi sao cac bat phuang inh sau nghiem diing voi moi x .
273
a) |x + l | + 2x^ - 2x + l > 0 b) x 2+ 2
7 ^
Huong dan gidi a) Tac6|x + l | > 0 , 2 x 2 - 2 x + l = ( x - l ) ^ + x 2 > 0 -
> 2
Suy ra |x + l | + 2x^ - 2 x + l > 0 DSng thuc xay ra khi va chi khi
x + l = 0
( x - l f + x 2 = 0 (v6 nghiem) Suy ra |x +1| + 2x^ - 2x + 1 > 0 v o i moi x .
' 'ay bat p h u o n g trinh nghiem dung voi moi x . ^' b) A p d u n g BDT cosi ta c6
•>2
x ^ + l i •^/77T = 2
Suy ra bat p h u o n g trinh nghiem dung voi m o i x .
Bai 4.59: Ban Binh giai bat p h u o n g trinh V X + 1(N/2X + 2 - 1 ) > 0 n h u sau Bat p h u o n g trinh t u o n g d u o n g voi
, V 2 x T 2 - l > 0 ô V 2 x + 2 > l o 2 x + 2 > l o x > - ^
2 ' ' xf V|y bat p h u o n g trinh c6 tap nghiem la ^ = [ - - ; +<ằ) •
Theo em ban Binh giai n h u vay diing hay sai? Neu sai hay sua lai ch'o dung.
Huong dan gidi Ban Binh da mSc sai lam 6 phep bien doi dau tien y Lai giai dung la:
V x T T ( V 2 x + 2 - l ) > 0 < ằ
<=>
• x = - l 2 x + 2 > l
x = - l
2
x + 1 - 0 V2x + 2- l > 0
x = - l V 2 ^ > 1
Vay bat p h u o n g trinh c6 tap nghiem la S = {-1} u 1 ^ - - ; . c o