ĐẠI SỐ LỚP 10 NÂNG CAO TIẾP THEO

Chuang

U.
fifll

f HUDTIG TRlnH

Bat ding thuc va bat phuong trinh la nhung khai niem ma
chung ta da lam quen 6 lop duoi. Chuong nay se hoan thien
hon cac khai niem do, dong thoi cung cap cho chiing ta nhung
ki^n thuc mdi nhu van de xet dau cua nhi thiic bac nhat va dau
cCia tam thuc bac hai. Chung co nhieu ung dung quan trong trong
viec giiii va bien lu^n cac phuong trinh va bat phuong trinh.
Chung ta can nam vung cac kien thuc do, dong thdi ren luyen
kl nang ap dung chiing de giai cac bai toan trong khuon kho
ciHa chuong tnnh.

103


t

/^_

BAT D A N G T H l / G
V A C H U N G M I N H B X T D A N G THtTC

1. On tap va bo sung tmh chat cua biit dang thurc

Gia sir a vab la hai sd thuc. Cac menh dd "a > b", "a < b", "a > b'\ "a < b
dugc ggi la nhung bdt dang thdc. .

Ciing nhu cac menh dd Idgic khac, mdt bdt dang thiic cd thd ddng hoac sai.
Chiing minh mdt bdt ddng thiic la chiing minh bdt dang thiic dd diing.
Dudi day la mdt sd tinh chdt da bidt ciia bdt dang thtfc.
a>b va b>c ^^ a>c
a>b <=> a + o b + c
Ndu c > 0 thi a > Z? «:> ac>be.
Ndu c<Othia>b<:^ac< be.

Tii dd ta cd cdc he qua sau :
a>b va c>d => a + c>b + d;
a + oh
<^a>b-c;
a>b>0 va c>d>0 ^^ aobd;
a>b>0 va neN* => a">b";
a>b>0 <:> yfa > 4b ;
a>b-^ \fa > ^fb.
Vi du 1. Khdng dung bang sd hoac may tinh, hay so sanh hai sd >/2 + >/3 vd 3.
Gidi. Gia suf V2 + \/3 < 3. Do hai vd ciia bdt dang thiic dd ddu dugng ndn
V2 + >^ <3 <^ (V2 + Sf<9

<^ 5 + 2V6<9

<^ 2yj6 < 4 <» V6 < 2 O 6 < 4, vd Ii.
vay V2 + V3 > 3.
104

•


Ndu A, B la nhiing bidu thiic chiia bidn thi "A > B" la mdt menh dd chiia bidn.

Chiing minh bdt dang thiic A>B (vdi dieu kien nao dd ciia cdc bidn), nghia la
chiing minh menh dd chiia bidn A>B diing vdi tdt ca cac gia tri cua cac bidn
(thoa man didu kien dd).
Tvi nay, ta quy udc : Khi ndi ta cd bdt dang thiic A > B (trong dd A va B la
nhiing bidu thute chiia bidn) ma khdng neu didu kien ddi vdi cdc bidn thi ta
bidu rang bdt dang thiic dd xay ra vdi mgi gia tri cua bidn thudc R.
Vi du 2. Chiing minh rang x^ > 2(x -1).
Gidi. x ^ > 2 ( x - l ) o x ^ > 2 x - 2 <» x ^ - 2 x + 2 > 0
o x ^ - 2 x + l + l>0<=> (x-l)^ + l > 0 .
Hidn nhidn (x - 1) + 1 > 0 vdi mgi x nen ta cd bdt dang thiic cdn chiing minh. n
Vi du 3. Chiing minh rang ndu a, b, c la dd dai ba canh cua mdt tam giac thi
ib + c - a)ic + a- b)ia + b - c) < abc.
Gidi. Ta cd cac bdt dang thiic hidn nhidn sau :
2

2

2

a > a -ib-c)

=ia-b + c)ia + b-c)

b^ >b^-ic-af

= ib-c + a)ib + c-a)

c^ >c^-ia-bf
= ic-a + b)ic + a- b).
Do a, b, c la dd ddi ba canh cua mdt tam giac ndn tdt ca cdc vd ciia cac bdt

dang thiic tren ddu duong. Nhan cac vd tuong ling cua ba bdt dang thiic tren,
ta duoc
a V c ^ >ib + c- afic + a- hfia + b-cf. '
Ldy can bdc hai cua hai vd, ta dugc bdt dang thiic cdn chiing minh.

D

2. Bat d^ng thurc ve gia tri tuyet doi
Tur dinh nghia gid tti ttiyet ddi, ta suy ra cac tinh chdt sau day.
a\< a< \a\ vdi mgi a G E.
|x |x >a<^x< -ahoacx>a (vdi a> 0).
-

105


Sau day la hai bdt ddng thiic quan ttgng khdc vd gia tri tuydt ddi (vidt dudi
dang bdt dang thiic kep).
a\ -\b\<\a+b\<\a\

+ \b\ (vdi mgi a, ft e M)

Ta chiing minh bdt dang thiic | a + 61 < | a | + 161. Thdt vdy
\a + b\<\a\+\b\

<^ ia + b)^
+2\ab\+b^

<:>a+2jab +

l3'
Bdt dang thiic cudi ciing ludn dung ndn ta cd bdt ddng thiic cdn chiing minh.

m l Sd dung bdt ding Ihdc vCfa ehCtng minh vd ding thdc \a\ = \a + b + i-b)
chdng minh bdt ding there \a\-\b\<\ a+b\.
3.

Bl(t Akng thurc giura trung binh cdng \k trung binh nh^n^^^
a) Ddi vdi hai sd' khong dm
Ta da bidt

a + b ,.
la trung binh cdng cua hai sd a va b. Khi avab khdng am

thi 4ab ggi Id trung binh nhdn cua chiing. Ta cd dinh li sau ddy.
DINHU
Vdi mgi a > 0, ft > 0 ta cd
a+b

> yfab.

Dang thiic xay ra khi va chi khi a = b.
Ndi cdch khdc, trung binh cdng cua hai sd khdng dm ldn han hodc bdng trung
binh nhdn cua chiing. Trung binh cdng cua hai sd'khdng dm bdng trung binh
nhdn cua chiing khi vd chi khi hai sddd bdng nhau.
(1) Ngudi ta con goi IJI bit ding tiiiic C6-si (Augustin-Louis Cauchy, 1789 - 1857).

106


Chitng minh. Vdi a > 0, ft > 0, ta cd
^^-4ab

=]-ia + b-2y[ab)

= -i4a

-Sf

>0.

Dodd
a+b

> yfab.

Dang thiic xay ra khi vd chi khi (Va - >/ft)^ = 0, hie la a = ft.

n

H2J Trong hinh 4.1, cho AH = a, BH = b. Hay tinh
cdc doan OD vd HC theo a vd b. Tir dd suy ra bat
ding thdc glOa trung binh edng vd trung binh nhdn
cQa a vd b.
Vi du 4. Chiing minh rang ndu a, ft, c Id ba sd
duong bdt ki thi
'i' a + b h + c

c+a
>6.
+
+
a
Gidi. Ta cd
a+b

+

b+c

c+a

a

t o o

c

Hinh 4.1

c

a

+ - + — + —+- + —
ft = —
c
c <3 a ft ft

c

a

+ —+—
'^^ft aJ U ftV
\a c)
^ ^ \a b ^ b c ^ \c a
>2.
+2.
+2.
=6.
Vft a
^c b ^ a c

a

H£ QUA
Niu hai so duang thay ddi nhung cd tdng khdng dd'i thi tich ciia
chiing ldn nhdt khi vd chi khi hai sddd bdng nhau.
Niu hai so duang thay ddi nhung cd tich khdng ddi thi tdng ciia
chung nhd nhdt khi vd chi khi hai sddd bdng nhau.
Chicng minh. Gia sir hai sd duong x vd y cd tdng x + y = S khdng ddi. Khi dd,
—=

— > .yfxy nen xy < — . Dang thiic xay ra khi vd chi khi x = y.

Do dd, tich xy dat gid tri ldn nhdt bang — khi va chi klii x = y.
107

Gia sir hai sd duang x va y cd tich xy = P khdng ddi. Khi dd
^ ^ ^ > 4xy = yfP nen x + y > 2>/P,
Dang thiic xay ra khi va chi khi x = y. Do dd, tdng x + y dat gid tri nhd nhdt
bdiig 2 v P Idii va chi khi x = y.

•

TJNGDIJNG
Trong tdt cd cdc hinh cha nhdt cd cung chu vi, hinh vudng cd dien
tich Ian nhdt.
Trong tdt cd cdc hinh ,chii nhdt cd cung diin tich, hinh vudng co
chu vi nhd nhdt.
3
Vi du 5. Tim gid tri nho nhdt cua ham s6fix) = x H— vdi x > 0.
X

Gidi. Do X > 0 nen ta cd fix) = x + — > 2 Jx.— = 2yi3 va
X

V

X

fix) = 2V3 <:^ X = - <:^ X = V3.
X

3

..

vay gia tri nhd nhdt cua ham s6fix) = x + — v d i x > 0 la/(>M) = 2v3.
b) Doi vdi ba sd Ichdng dm
Ta da bidt

flf + ft + C

la trung binh cdng cua ba sd a, ft, c. Ta ggi Vflftc la

trung binh nhan cua ba sd dd. Ngudi ta ciing chiing minh dugc kdt qua tuong
tu dinh Ii tren cho trudng hgp ba sd khdng am.
Vdi mgi a > 0, ft > 0, c > 0, ta cd
3
Dang thiic xay ra khi va chi khi a = b = c.

Ndi each khdc, trung binh cdng cOa ba sd khdng dm ldn han hodc bdng trung
binh nhdn cua chiing. Trung binh cdng ciia ba sd'khdng dm bdng trung binh
nhdn cua chiing khi vd chikhi ba sddd bdng nhau.
108


Vi du 6. Chiing minh rang ndu a,ft,c la ba sd duong thi
1 1 1
ia + b + c) — + - + - >9.
a b cj
Khi ndo xay ra dang thiic ?
Gidi. Vi a,b,c la ba sd duong nen
a + b + c > 3 ¥abc (dang thiic xay ra khi va chi khi a =ft= c) va

i,i4>33-i-' dang thiic xay ra khi va chi khi — = — = —

a b c

Do dd

V abc

a b c

I I I
(a +ft+ c) —+ - + - >3Vaftc.3 3 / — =9.
a b c
\ abc
a =b=c

Dang thiic xay ra khi va chi khi

1-1-1
a b c

Vdy dang thiic xay ra Idii va chi khi a = b = c.

a

H3| Phat biSu kd't qua tuong tuhd qud d phin a) cho trudng hgp ba so duang.

Cau hoi va bdi tap
1. Oiling minh rang, ndu a > ft va aft > 0 thi — < —
a ft
2. Chting minh rdng nira chu vi cua mdt tam gidc ldn hon dd ddi mdi canh cua tam
gidc dd.

3. Chiing minh rang a^ + b^ + c^ > ab + be + ca vdi mgi sd thuc a, ft, c.
Ding thiic xay ra khi va chi khi a = ft = c.
4. Hay so sdnh cdc kdt qua sau day :
a) V2000 + V2005 vd V2002 + V2003 (khdng diing bang sd hoac may tinh);
b) yla + 2 + yJa + A va yfa + yfa + 6 ia > 0).
109


5.

1 1
4
Chiing minh rdng, ndu a > 0 v a f t > 0 t h i — + — >
a ft a + b

6.

Chiing minh rdng, neu a > 0 va ft > 0 thi a^ + ft^ > abia + ft). Dang thiic
xay ra khi nao ?

7.

a) Chiing minh rdng a^ + aft + ft^ > 0 vdi mgi sd thuc a, ft.
b) Chiing minh rang vdi hai sd thuc a, ft tuy y, ta cd a^ +b^ > a^b + ab^

8.

Chiing minh rdng, ndu a, ft, c la dd dai cdc canh cua mdt tam gidc thi
a^ +b^ +c^ < 2(aft + bc + ca).

9. Chiing minh rang, ndu a > 0 vd ft > 0 thi
a +b

a^ +b^

10. a) Chiing minh rang, ndu x > y > 0 thi

a^ +b^

^ >^ y
1+x
1+y

b) Chiing minh rdng ddi vdi hai sd tuy y a, ft, ta cd -^—!—<
l + |a-ft|

1^1 .^ I I
l + |a| l + |ft|

11. Chiing minh rang :
a) Ndu a, ft la hai sd ciing ddu thi —+— > 2 ;
ft a
b) Ndu a, ft la hai sd trdi ddu thi - + - < - 2 .
ft a
12. Tun gid tri ldn nhdt va gia tti nhd nhdt ciia ham sd fix) = (x + 3)(5-x) vdi
-32
13. Tim gia tri nhd nhdt cua ham sd fix) = x +
vdi x > 1.
x-l

110


Bdi doc them
BAT DANG THQC BU-NHI-A-C6P-XKI(^)

1. Bat dang thiic Bu-nhi-a-cop-xki doi vdi hai cap sdthi/c
Vdi hai cap sd thUc (a,b) va (x,y) ta cd
iax + byf < ia' + b\x- + y\
Ding thiic xdy ra khi va chi khi ay = bx.
Chdna minh.
D l dang chdng minh ding thdc sau :
iax + by-f + iay - bxf = ia^ + b\x^ + y \
Udt Ichac, do iay - bxf > 0 nen
iax + byf + iay - bxf > iax + byf.
lit do suy ra bat ding thdc can chdng minh.
Ding thdc x^y ra lChOv. Khi xy ?t 0, dieu lX

y

2. Bat ding thiic Bu-nhi-a-cop-xki ddi vdi hai bg ba sdth^c
C6 the chdng minh ket qud sau :
Vdi hai bd ba sd thuc (ai, a2, a;}),(bi, b2, b^, ta cd
(ai&l + ^2^2 + «3''3)^ ^ ('^i^ + ^ + ^3)(bi + bl + bl)
Neufc,bob'ii^O thi ding thdc xdy ra khi v^ chi khi — = ^
6,

= ^•

foj

foj

VI dy. Chiing minh rang neu a^ + 21? + 9c^ = 3 thi a + 26 + 9c < 6.
Gi^i. Ta c6 (a + 26 + 9cf = (a. 1 + >/2 6. >/2 + 3c.3)^ <
< \
+ 3^] = 12 (fl2 + 2b^ + 9c2) = 36.

Vivdy a + 26 + 9c<6.

D

(1) Viktor Yakovlevlch Bunyakovsky (1804 - 1889), nha toan hoc Nga.

Ill


Luyen tap
14. Chiing minh rang ndu a, ft, c la ba sd duang thi
4

i4

4

— + — + — > 3aftc.

a^
b
c
15. Mdt khach hang ddn mdt cxta hang ban hoa qua mua 2 kg cam da ydu cdu
cdn hai ldn. Ldn ddu, ngudi ban hang dat qua cdn 1 kg Idn dia cdn bdn phai
va dat cam len dia can ben trai cho ddn khi can thdng bdng vd ldn sau, dat
qua cdn 1 kg len dia can ben trai va dat cam len dia can ben phai cho ddn khi
can thang bdng. Ndu cdi cdn dia dd khdng chinh xdc (do hai cdnh tay don
dai, ngdn khdc nhau) nhung qua can la dung 1kg thi khach hang cd mua
dugc diing 2 kg cam hay khdng ? Vi sao ?
16. Chiing minh rang vdi mgi sd nguyen duong n, ta cd :
. 1 1 1
1
,
a) — + — +
+ ... +
< 1;
1.2 2.3 3.4
nin +1)
Hudng ddn. Vidt — = 1
; — =
1.2
2 2.3 2

,...
3

b) "Y + ^ + ^ + ... + ^ < 2.
r
2^ 3^

n^
17. Tim gid tri ldn nhdt va gia tri nhd nhdt ciia bidu thdc
A = V x - 1 + V4-X.
18. Chiing minh rang vdi mgi sd thuc a, ft va c, ta cd
(a + ft + cf < 3ia^ + ft2 + c \
19. Chiing minh rang ndu a,ft,c, d la bdn sd khdng dm thi a+b+c+d

A4

> abed.

20. Chiing minh riyng :
a) Neu x^ + y^ = 1 thi I x + y I < >/2 ; b) Ndu 4x - 3y = 15 thi x^ + y^ > 9.
112


DAI CLfONG v £ BXT PHLfONG TRINH

1. Khai niem bat phuong trinh mot an
DINH NGHIA
Cho hai hdm sdy =f(x) vdy = g(x) cd tap xdc dinh ldn lu0 la S)^
vd2)^.£>a7 2) = ® ^ n %
Menh di chica biin cd mdt trong cdc dang fix) < g(x),
fix) > g(x),f(x) < g(x),f(x) > g(x) dugc ggi la bdt phuang
trinh mgt dn ; x ggi la dn sd (hay dn) vd 3) ggi la tap xdc dinh
ciia bdt phuang trinh dd.
Sd XQ e 2) ggi la mdt nghiem cua bdt phuang trinh fix) < g(x)
niufixf)) < g(xQ) la menh di diing.
Khdi niem "nghiem" ciing dugc dinh nghia tuong tu cho cdc bdt phuong trinh dang
fix) > gix), fix) < gix) va fix) > gix).

Gidi mdt bdt phuong trinh la tim tdt ca cdc nghiem (hay tim /op nghiem) cha
bdt phuang trinh dd.
CHU Y
Trong thuc hanh, ta khdng cdn vidt rd tap xac dinh 2) cua bdt
phuang ttinh ma chi cdn neu didu kidn dd x e 9). Dieu kien dd ggi
la diiu kiin xdc dinh cua bdt phuang trinh, ggi tdt la dieu kien cua
bdt phuang trinh.
m l Bi0u dien tdp nghidm cda mdi bat phuang trinh sau bai cac kl hieu khoang
ho$c doein:

a)-0,5x>2;
8.0/!USdlO(NC)-ST^

b) \x\ < 1.

113


Dudi ddy, chiing ta chi ndi tdi bdt phuong ttinh dang fix) < gix). Ddi vdi cac bdt
phuong ttinh dang fix) > gix), fix) < gix) vafix) > gix), ta ciing cd cdc kdt qua
tuong tu.
2.

Bat phuong trinh tirofng dirong
DINH NGHIA
Hai bdt phuang trinh (ciing dn) dugc ggi la tuang duang niu
chiing cd ciing tap nghiem.
Niufiix) < giix) tuang duang vdif2ix) < g2ix) thi tdviit
fiix) < giix) o /2(x) < g2ix).
H2| Cdc khang djnh sau ddy ddng hay sai ? Vi sao ?

a) x+ Vx-2 > Vx-2 <=>X> 0 ;

b) i4x^f

<\

<^x-\<\.

CHUY
Khi mudn nhdn manh hai bdt phuang trinh cd cung tdp xac dinh 3)
(hay cd cung didu kidn xac djnh ma ta ciing ki hieu la 2)) va tuong
duong vdi nhau, ta ndi:
- Hai bdt phuong trinh tuong duang tren 3), hoac
- Vdi dieu Iden 3), hai bdt phuang trinh la tuofng duang vdi nhau.
Vl du 1. Vdi didu kien x > 2, ta cd
3.

>l<:>l>x-2.

D

Bien ddi tuong duong cac bat phuong trinh
Ciing nhu vdi phuang trinh, d ddy chiing ta quan tdm ddn cac phep bidp ddi
khdng lam thay ddi tdp jighidm cua bdt phuofng trinh. Ta ggi chiing la cac jdiep
bien doi tuang duang. Phep bie'n doi tuang duang biin mdt bdt phuang trinh
thdnh mdt bdi phuang trinh tuang duang vdi nd. Chang ban, vide thuc hidn cac
phep bidn ddi ddng nhdt d mdi vd cua mdt bdt phuang trinh va giii nguydn tdp
xac dinh cua nd Id mdt phep bidn ddi tuofng duong.

114

8. BifUSdlO(NC)-8T-B


Dudi ddy la dinh Ii vd mdt sd phep bidn ddi tuong duang thudng diing. Cac ham
sd ndi ttong dinh Ii nay ddu dugc cho bdi bidu thiic.
DINH U
Cho bdt phuang trinh fix) la mdt hdm so xdc dinh trin 2).
Khi dd, trin 3), bdt phucmg trinh fix) < gix) tuang duang vdi
mdi bdt phuang trinh :
I) fix) + hix) < gix) + hix);
2)/(x)/i(x) < gix)hix) niu hix) > 0 vdi mgi x e 2);
3)fix)hix) > gix)hix) niu hix) < 0 vdi mgi x s9).
Chicng minh. Sau day, ta chi chiing minh ket luan 3). Cac kdt luan khdc ciing
dugc chiing minh tuong tu.
Ndu XQ thudc 2) thi /(XQ), ^(XQ) va /i(xo) la cdc gia tri xdc dinh bang sd,
hon niia, vi ft(x) ludn dm ndn /i(xo) < 0. Do dd, dp dung tinh chdt cua bdt
ddng thiic sd, ta cd
fiXo) < giXo) <:> fiXo)hiXQ) > g(Xo)/l(Xo).

Tii dd suy ra rdng hai bdt phuang ttinh cd ciing tap nghidm, nghia la chiing
tuong duong vdi nhau.
D
Vidu2
a) Bdt phuong ttinh Vx > -2 tuong duong vdi bdt phuong ttinh

4x-4x>-2-4x.
b) Bdt phudng ttinh x > - 2 khdng ttrong duong vdi bdt phuong trinh

x-Vx>-2-Vx.

D

H3J ChOng minh cdc khing dinh trong vi du 2.
H4| Cdc khing djnh sau ddy dOng hay sai ?Visao?

a)x+ — <1 + — <:>xX

x

x-l
115


HE QUA
Cho bdt phuang trinh fix) < gix) cd tap xdc dinh 2).
1) Quy tdc ndng lin luy thiCa bdc ba
fix)2) Quy tdc ndng lin luy thuca bdc hai
Niu fix) vd gix) khdng dm vdi mgi x thudc 2) thi
fix)Tuofng.tir, ta ciing cd quy tdc nang len luy thiia bac le va nang len luy thiia bac chan.

H5| Giai bat phuang trinh sau ddy (bang each binh phuong hai vd), giai thich
phep bien ddi tuong dUOng da thuc hien :
|x+l|

Cau hoi va bai tap
21. Mdt ban lap luan nhu sau : Do hai vd cua bdt phuong trinh v x - 1 < |x|
ludn khdng am ndn binh phuong hai vd, ta dugc bdt phuong trinh tuang duong
X - 1 < X . Theo em, lap luan tten cd dung khdng ? Vi sao ?
22. Tim didu kidn xdc dinh rdi suy ra tap nghiem cua mdi bdt phuofng trinh sau :
a) 4x > yT-x ;
c) X +

x-3

b) Vx - 3 < 1 + Vx - 3 ;

>2+

x-3

;

d)

.
<
Vx-2
Vx-2

23. Trong hai bdt phuong trinh sau ddy, bdt phuang trinh nao tuofng duong vdi bdt
phuang trinh 2x - 1 > 0 :
2x-l + — ^ > - ^
x-3

x-3

vd 2 x - l

^ > —

x+3

^

x+3

24. Trong bdn cap bdt phuang trinh sau ddy, hay chgn ra cdc cap bdt phuong trinh
tuong duong (ndu cd):
a) X - 2 > 0 va x^(x - 2) < 0 ;
c) X - 2 < 0 vax^(x- 2) < 0 ;
116

b) x - 2 < 0 vd x^(x - 2) > 0 ;
d) x - 2 > 0 vax^(x- 2) > 0.


PHLfONG TRINH vA Hfi BXT
TRINH BAc N H X T M 6 T X N
BXT

PHLTONG

Trudc day, chiing ta da lam quen vdi bdt phuang trinh bdc nhdt mot dn. Dd la
bdt phuang trinh cd mdt ttong cdc dang ax + ft<0, ax + ft<0, ax + ft>0,

ox +ft> 0, trong dd a va ft la hai sd cho trudc vdi a ^t 0, x la dn.
HI] Cho bit phuang trinh mx < mim + 1).
a) Giii bdt phuong trinh vdi m = 2.
b) Giai bdt phuong trinh vdi m = —\/2.
Nhu vdy, ndu a va ft la nhiing bidu thiic chiia tham sd thi tap nghiem ciia bdt
phuong trinh phu thudc vao tham sd dd. Vide tim tap nghiem ciia mdt bdt
phuong trinh tu^ theo cdc gid tri cua tham sd ggi Id gidi vd biin ludn bdt
phuang trinh dd.
Dudi ddy, chiing ta chu ydu ndi vd each giai va bien luan bdt phuong trinh
dang ax + ft < 0. Ddi vdi cac bdt phuong trinh dang cdn lai, each giai cung
tuong tu.
1. Giai va bien lu^n bat phuong trinh dang ax + b<0
Kit qua giai vd bidn luan bdt phuong trinh
ax +
dugc ndu ttong bang sau day.

ft,,

ft<0

,

(1)

r

b]

1) Ndu a > 0 thi (1) <» x < — . Vay tap nghidm ciia (1) la 5 =
V

aJ
a
( b
\
2) Ndu a < 0 thi (1) <=> X > — . Vdy tdp nghiem cua (1) la S =
— ;+oo
a
\ a
J
3) Ndu a = 0 thi (1) o Ox <-ft. Do dd :
- Bdt phuang ttinh (1) vd nghiem (S = 0 ) ndu ft > 0 ;
- Bdt phuang ttinh (1) nghiem dung vdi mgi x (S = R) ndu ft < 0.
117


CHUY
Viec bidu didn cdc tap nghidm tten true sd se rdt cd ich sau ndy,
Chang ban, phdn khdng bi gach d tten hinh 4.2 bidu didn tdj
nghiem cua (1) vdi a > 0.

Hinh 4.2

Vi du 1. Giai va bien ludn bdt phuofng trinh
mx + l>x + m .

(2)

Gidi. Bdt phuofng trinh (2) tuong duong vdi
im-l)x>m^-l.

(3)

Ta cd
m^ - 1
1) Ndu m > l t h i / n - l > 0 nen (3) «> x > ——- <» x > m + 1.
m- -1
m^ - 1
<» x < m + 1.
2) Ndu m < l t h i m - l < 0 nen (3) <=> x <
m-l
3) NduOT= 1 thi bdt phuong trinh ttd thanh Ox > 0 nen nd vd nghidm.
Kit ludn : -Ndu w > 1 thi tap nghiem cua (2) la 5 = (m + 1 ; +oo).
- Ndu m- Ndu m = 1 thi tap nghiem cua (2) la ,S = 0 .

D

H2| TCrke't qua tren, hay suy ra tdp nghiem cQa bdt phuang tnnh

mx+i >x + m :

V i d u 2 . Giai va bien ludn bdt phuang trinh
•

2mx >x + Am-3.

(4)

Gidi. Bdt phuang ttinh (4) ttiong duang vdi
(2m-l)x>4m-3.

118

(5)


1
Am-3
l ) N d u / n > - thi2m-l>0nen(5)<i>x>-^^^—^•
2
2m-l
2) Ndu m < - thi 2m - 1 < 0 nen (5) o x <
•
2
2m-l
3) Ndu m= — thi (5) ttd thanh Ox > - 1 , bdi vay nd nghidm dung vdi mgi x.
Kit ludn : - Ndu m > — thi tdp nghidm ciia (4) laS =

Am-3

^

;+co
2m - 1
y
r
4
m
3
- Ndu m< — thi tap nghiem cua (4) la 5 = - 0 0
2m-1

- Ndu m = — thi tdp nghiem cua (4) la 5 = ^

D

2. Giai he bat phuong trinh b$c nhsi't mot an
Tuong tu nhu he phuang trinh, tap nghiem cua mdt hd bdt phuong trinh la
giao ciia tdt ca cac tap nghiem ciia cac bdt phuong trinh ttong he.
Do dd,
Mudn gidi hi bdt phuang trinh mot dn, ta gidi ticng bdt phuang
trinh ciia hi rdi ldy giao cua cdc tap nghiim thu dugc.
Vl du 3. Giai he bdt phuang trinh
(I)

'3x-5<0
< 2x + 3 > 0
X + 1 > 0.

(6)
(7)
(8)

Gidi. Giai ldn lugt ttoig bdt phuong trinh cua he, ta dugc :
Tap nghiem cua (6) la S^ =
Tdp nghiem ciia (7) la Sj

—00

3
— :+oo

2

T$p nghiem ciia (8) la 53 = ( - 1 ; +«)
119


vay tap nghiem cua he bdt phuong trinh da cho Id
S = Si n 52 n 53 =

f

5]

-1

V

Ta cung cd the trinh bay loi giai vi du 3 nhu sau :

(I)

«

3
^ 3

,

X>

•

3J

n

5

<::>-l < X < —

x>-l

2

3

Tap nghiem cua he bdt phuong trinh da cho la
/
S=

-1
V

5]

n

3J

CHU Y

Dd de xac dinh tap nghiem S, ta bidu didn cdc tdp nghiem tten true
sd bdng each gach di cac didm (phdn) Ichdng thudc tdp nghidm cua
tiing bdt phuang trinh ttong hd, phdn cdn lai se bidu didn tdp
nghiem cdn tim (h.4.3).
-1

-1
Hinh 4.3

H3| 77m cac gid tri cOa x di ddng thdi xay ra hai ding thdc:
|3x + 2| = 3x + 2 v a | 2 x - 5 | = 5-2x.

Hu&ng ddn.\A\ = A<:^A>Ova\B\ = -B<:>BVi du 4. Vdi gid tri nao ciia m thi he bdt phuang trinh sau cd nghidm ?
rx + m < 0
\ - x + 3 < 0.

(9)
(10)

Gidi. Ta cd (9) o x < -m. Tdp nghiem ciia (9) la (-00 ; -m\.
(10) <:> X > 3. Tdp nghidm ciia (10) la (3 ; +00).
Vdy tap nghiem ciia he Id S = (-00 ; -m] n"(3 ; +00). Hd cd nghifim khi vd
chi khi S ?!^ 0 , hie la 3 < - m hay m < - 3 .
•
120


Cau lioi va bai tap
25. Giai cdc bdt phuofng trinh :

X x+2
a) —
X + 1> X + 3 ;

3x + 5 . ^ X + 2
b ) - ^ - l < _ - . . ;

c) (1 - V2)x < 3 - 2V2 ;

d) (x + V3)2 > (X - V3)2 + 2.

26. Giai va bien luan cac bdt phuang ttinh
a) m(x - m) < X - 1 ;
b) mx + 6 > 2x + 3m ;
c) (x + 1)A: + X < 3x + 4 ;
d)(a+l)x + a + 3 > 4 x + l .
27. Giai cdc hd bdt phuofng ttinh :
^ r5x - 2 > 4x + 5
12x + 1 > 3x + 4
a) i
b)
[5x - 4 < X + 2 ;
!5x + 3 > 8 x - 9 .

Luyen tap
28. Giai vd bien luan cac bdt phuofng trinh
a)m(x-m)>2(4-x);
c ) % - l ) + 4x>5;
29. Giai cdc he bdt phuang trinh :
'5x + 2

>4-x
3
a)
6-5x
< 3x+.l;
13
4x-5
7
c)
3x + 8

> 2x - 5 ;

b)3x + m^>m(x + 3);
d)ft(x-l)<2-x.

(1 - x)^ > 5 + 3x + x^
b)

(x + 2)^ < x^ + 6x2 - 7x - 5 ;
X - 1 < 2x - 3
d) 3x < X + 5
5-3x

30. Tim cdc gia tri cua m dd mdi he bdt phuang trinh sau cd nghiem :
^ r 3 x - 2 >.-4x + 5
rx-2<0
^) 1

b) -^
[3x + m + 2 < 0 ;
•
[m + x > l .
31. Tim cdc gid tri cua m dd mdi hd bdt phuang trinh sau vd nghidm :
a)

f2x + 7 < 8x - 1
[-2x + m + 5 > 0 ;

b)

| ( x - 3 f >x2+7x + l
2m - 5x < 8.
121


DAU CUA NHI THLfC BAC N H A T

1. Nhi thurc b$c nhat va d^u cua no

Nhidu bai toan ddn den viec xet xem mdt bidu thvccfix)da cho nhan gid tri am
(hoac duang) vdi nhiing gia tri nao cha x. Ta ggi vide lani dd la xet ddu cda
biiu thdc fix). Dudi day, ta se tim bidu vd nhi thdc bac nhdt va ddu cua nd.
a) Nhi thurc b^c nhat
DINH NGHiA

Nhi thdc bdc nhdt (dd'i vdi x) la biiu thdc dang ax + b, trong ddava
ft la hai so cho trudc vdi a ^ 0.
Ta da bidt, phuofng trinh ax + ft = 0 (a 9^ 0) cd mdt nghidm duy nhdt XQ= — .

a
Nghidm dd cung dugc ggi la nghiem ciia nhi thicc bdc nhdt fix) = ax + b. No
cd vai trd rdt quan trgng trong viec xet ddu ciia nhi thiic bac nh^t fix).
b) Dau cua nhi thurc bac nhdt
Dat XQ = — , ta vidt nhi thdc bdc nh^t fix) = ax + b nhu sau
a

b^
fix) = ax + b= aL
*x + — = a(x-Xo).
V
uj
Khi X > XQ thi X - XQ > 0 nen ddu cua a(x - XQ) trung vdi ddu cua a.
Khi X < XQ thi X - Xo < 0 nen ddu cua a(x - XQ) ttdi vdi ddu cua a.
122


Ttt dd ta cd
DINH U (vd dau cua nhj thdc bac nhat)
Nhi thicc bdc nhdt fix) = ax + b cUng ddu vdi hi so a khi x ldn
han nghiim vd trdi ddu vdi hi sd'a khi x nhd hem nghiim ciia nd.
Kdt qua cua dinh li nay dugc tdm tdt trong bang sau
—00

trdi ddu vdi a

fix) -ax + b

C3iang ban nhi thiic fix) = -x + 1,5
cd he sd a = - 1 vd nghiem XQ = 1,5.

Dodd, ddu cua nd dugc cho ttong
bang sau
1,5

—00

- x + 1,5

+

+00

XQ

+00

0

0

cung ddu vdi a

)'•

y\

X)

i
0

/
a<0

a>0

Nhi thiic da cho duang khi x < 1,5
vaamkhix> 1,5.

'^

Hinh 4.4

H1| Hdy giai thich bhng dd thi (h.4.4) cdc ket qud cOa djnh II trdn.
2. Mot so umg dung
a) Giai bdt phuong trinh tich
Ta xdt cdc bdt phuong ttinh cd thd dua vd mdt ttong cdc dang P(x) > 0, P(x) > 0,
P(x) < 0, Pix) < 0 vdi Pix) Id tich cua nhiing nhi thiic bac nhdt.
Vi du 1. Giai bdt phuong ttinh
(x-3)(x+l)(2-3x)>0.

(1)

Cdch gidi. Di giai bdt phuang trinh (1), ta lap bang xet ddu vd ttdi ciia (1).
Dat Pix) = (x - 3)(x + 1)(2 - 3x).
- Gidi phuang ttinh P(x) = 0, ta dugc
(x - 3)(x + 1)(2 - 3x) = 0 o X = 3 hoac X = - 1 hoac X = - •
123


- Sap xdp cac gia tri tim dugc cua x theo thii tu tang : - 1 , —. 3. Ba sd nay chia

true sd thanh bdn khoang. Ta xac dinh ddu cua Pix) ttdn tiing khoang bang cdch
Idp bang sau ddy ggi la bdng xet ddu cua Pix).
X

—00

2
—

—1

-xJ

+00

3
x-3

-

x+1

-

• 2 - 3x

+

Pix)

+

0

-

-

+

+

0

+

+ . 0
0

-

0

+

-

+

0

Trong bang xet ddu, hang ttdn ciing ghi lai bdn khoang dugc xet cua true sd,
ba hang tidp theo ghi ddu cua cdc nhan td bac nhdt trdn mdi khoang (dua vao
dinh li vd ddu cua nhi thiic bac nhdt) ; hang cudi ghi ddu cua Pix) ttdn mdi
khoang bang cdch ldy "tich" cua cac ddu cung cdt d ba hang ttdn.
Dua vdo bang xet ddu, ta cd tdp nghiem cua bdt phuong trinh (1) la
5 = (-oo;-l)u

ir'l-

D

b) Giai bat phuong trinh chura an d mdu
O day, ta chi xet cac bdt phuang trinh cd thd dua ve mdt ttong cdc dang
S ^ < ^' 5 4 - ^ ' 5 4 - 0 ' ^ > 0 , ttong dd ^(x) vd Qix) Id tich cua
Qix)
Qix)
Qix)
Qix)
nhiing nhi thiic bdc nhdt. Di giai cac bdt phuofng tririh nhu vay, ta lap bang xet
ddu cua phan thdc Pjx) Khi lap bang xet ddu, nhd rang phai ghi tdt ca cac
Qix)
nghiem cua hai da thiic Pix) va Qix) len ttiic sd. Trong hang cudi, tai nhiing
didm ma Qix) = 0, ta diing ki bidu || dd chi tai dd bdt phuang trinh da cho
khdng xac dinh.
Vi du 2. Giai bdt phuang trinh
3
x-2
124

<

2x - 1

(2)


Gidi. Ta cd
<^

<0
x-2
2x - 1
«> 3(2x - 1) - 5(x - 2) < 0
(x - 2)(2x - 1)
x+7
<0.
(x - 2)(2x - 1)

(2)

(3)

Bang xet ddu vd trdi cua (3):
X

-00

—7

—

2

+00

2x+7

-

x-2

0

+

+

-

- -

-

2x-l

-

-

0

+

Vdttdi

-

+

II

-

0

Tir dd suy ra tap nghidm cua (2) la 5 = (-oo ; -7] u | - ; 2

+
0

+
+

II

+
D

c) Giai phuong trinh, bat phuong trinh chura ^n trong ddu gia tri tuyet ddi

Mdt ttong nhiing each giai phuang trinh hay bdt phuong trinh chiia dn ttong
ddu gia tri tuydt dd'i la sir dung dinh nghia dd khd ddu gid tti tuydt ddi. Ta
thudng phai xet phuofng trinh hay bdt phuong trinh ttong nhidu khoang (doan,
nira khoang) khdc nhau, tten do mdi bidu thiic nam ttong ddu gid tri tuyet ddi
ddu cd mdt ddu xdc dinh. Sau day la mdt vi du don gian.
Vi du 3. Giai bdt phuang ttinh
12x - 11 < 3x + 5.
Gidi

(4)

1) Vdi X < - , ta cd
4
(4) <» 1 - 2x < 3x + 5 o 5x > ^ o X > —-•
1
4
1
Kdt hgp vdi didu kidn x < - , ta dugc —- < x < - . Vdy tdp cdc nghiem thoa
-^
z
/ 4 3
1
man didu kien dang xet la khoang
I 5 2J
125


2) Vdi X > - , ta cd
2
(4) <» 2x - 1 < 3x + 5 <:> X > - 6 .

Kdt hgp vdi didu kien x > —, ta dugc x > —. Vdy tdp cdc nghidm thoa man
didu kien dang xdt la niia khoang

!;+»

Tdm lai, tap nghiem cua bdt phuong trinh (4) Id
S=

A j_
u 1;+.
"5'2

-|;+oo|.

Cau lioi va bai tap
32. Ldp bang xdt ddu ciia cdc bidu thdc :
4-3x
a)
2x + l
c) x(x - 2) (3 - x);

b)ld)

2-x
3x-2

x(x - 3)^
(x - 5)(1 - x)

33. Phdn tich cac da thdc sau thdnh nhdn td bac nhdt rdi xdt ddu :

a) -X + X + 6 ;

b) 2x^ - (2 + V3)x + Vs.

34. Giai cdc bdt phuong trinh :

(3-.)U-2)^p.

K^
b)

X+ 1

c) | 2 x - V 2 | + | V 2 - x | > 3 x - 2 ;

3
. 5
>
1 - X 2x + 1

d)|(V2-V3)x+l|
35. Giai cdc he bdt phuang trinh :
'(x-3)(V2-x)>0
a) i 4x - 3
126

b)
< x + 3;

2 < 1
2x - 1 3 - X

Ixl < 1.


Luyen tap
36. Giai vd bien luan cdc bdt phuong trinh :
a)mx + 4 > 2 x + m ;

b) 2mx + 1 > X + 4m ;

c) x(m - 1) < m - 1;

d) 2(m + l)x < (m + 1)^ (x - 1).

37. Giai cac bdt phuong trinh :
3-2x

<0 ;

a) (-V3x + 2)(x + l)(4x - 5) > 0 ;

b)

, -3x +1 ^ .
c)
< -2
2x + l

,. x + 2 ^ x - 2
d)
<
3x + 1 2x - 1

(3x - l)(x - 4)

38. Giai va bien luan cdc bdt phuong trinh :

b) ^.-\X - 2m + 1

a) (2x - V2)(x - m) > 0 ;

39. Hm nghiem nguyen cua mdi hd bdt phuong trinh sau :

a)

6x + - > 4x + 7
7
8x + 3

< 2x + 25;

15x - 2 > 2x + 3
b)
3x-14
2(x - 4) <

40. Giai cdc phuofng trinh vd bdt phuong trinh :

a)|x+l| + | x - l | = 4;

b)

2x - 1

1

> —

(x + l)(x - 2)

2

41. Giai vd bien luan cdc he bdt^phuong ttinh
a)

f(x - V5)(V7 - 2x) > 0
IX - m < 0 ;

2
5
<
2x - 1
b) x - l
X - m > 0.

127