^ho thue xe chi c6 10 xe hieu M I T S U B I S H I va 9 xe hieu F O R D . Mgt chiec xe
•ii?u M I T S U B I S H I c6 the cho 20 nguoi va 0,6 tan hang. Mgt chiec xe hieu
^ O R D CO the cho 10 nguai va 1,5 tan hang. Tien thue mgt xe hi?u
M I T S U B I S H I la 4 trl^u dong, mgt xe hieu F O R D la 3 trieu dong. Hoi phai
*hue bao nhieu xe moi loai de chi phi thap nhat?
Phan loai va phuang phdp giat tJat so KT
Huang dan giai
G Q I X, y (x, y e N ) Ian l u o t la so xe loai MITSUBISHI, loai FORD can thu§
T u bai toan ta duoc he bat p h u a n g trinh
CtyTNHHMTV DVVll KItang Vi?t
0 < x < 10 0 < y < 9 20x + lOy > 140 0,6x + l , 5 y > 9
0 < X < 10 0 < y < 9
" 2x + y > 14
2x + 5 y > 3 0 I Tong chi p h i T ( x , y ) = 4x + 3y (trieu dong)
Bai toan tro thanh la t i m x, y nguyen khong am thoa man h^ (*) sao cho T(X, y ) nho nha't.
T u do ta can thue 5 xe hieu MITSUBISHI va 4 xe hieu FORD thi chi p h i van tai la thap nhat. ^-H- <".-.•
Bai 4.63: N h a n d i p tet Trung Thu, X i nghiep san xuat banh Trang muon san xuat hai loai banh: Dau xanh, Banh deo nhan dau xanh. De san xuat hai loai banh nay, X i nghiep can: D u o n g , Dau, Bot, Trung, M u t , ... Gia su so duong CO the chuan bj dugc la 300kg, dau la 200kg, cac nguyen lieu khac bao nhieu Cling CO. San xuat mot cai banh dau xanh can 0,06kg d u o n g , 0,08kg dau \ cho lai 2 ngan dong. San xuat mot cai banh deo can 0,07kg duong, 0,04k- dau va cho lai 1,8 ngan dong.
Can lap kehoach de san xuat m o i loai banh bao nhieu cai de khong bi don.;
ve duong, dau va tong so lai thu dugc la Ian nhat (neu san xuat bao nhii u cung ban het)?
Huong dan giai
Goi X, y Ian l u g t la so cai banh Dau xanh, banh Deo (x, y e N ).
•6x + 7y < 30000
§4. B A T P H U O N G T R I N H
V A H E B A T P H U O N G T R I N H B A G N H A T M O T A N
^OM T A T L Y THU Y E T .
gi^i va bi^n luan bat phuang trinh dang ax + b < 0 . Giai bat p h u a n g trinh dang ax + b < 0 (1)
pjgu a = 0 thi bat p h u a n g trinh c6 dang 0.x + b < 0
. Vai b < 0 thi tap nghiem BPT la S = R ^ H O i i / i t ^ i i p i 1 , Voi b > 0 thi tap nghiem BPT la S = 0
ivje'u a > 0 thi (1) <=> x < — suy ra tap nghiem la S = ' b^ -co;
a) Me'u a < 0 thi (1) o x > - - suy ra tap nghiem la S = ^ b — ;+oo
a
Bai toan tro thanh t i m so tu' nhien x, y thoa man he fx = 625 ' sao cho L = 2x + l , 8 y Ion nhat. T u do ta c6 •
y = 3750
2x + y < 5000
thi L = 2x + l,8y^^'' gia trj Ian nhat.
Vay can 625 banh dau xanh va 3750 banh deo thi lai Ian nhat.
xmm r
Cac bat p h u a n g trinh dang ax + b > 0, ax + b < 0, ax + b > 0 dugc giai hoan toan tuong t u
2, H? bat phuang trinh bac nhat mot an " •
De giai he bat p h u o n g trinh bac nhat mot an ta giai t u n g bat p h u o n g trinh cua he bat p h u a n g trinh. K h i do tap nghiem ciia he ba't p h u a n g trinh la giao cua cac tap nghiem tung bat p h u a n g trinh.
CAC DANG TOAN VA P H l / a N G PHAP G I A I .
DANG T O A N 1: GIAI BAT PHUONG TRINH DANG ax + b < Q.
^ 1. C A C V f D U M I N H H O A
^' 1: Giai va bien luan bat p h u a n g trinh sau.
+ 6 < 2x + 3m b) (x + m ) m + x > 3 x + 4 ni^ + 9)x + 3 > m ( l - 6 x ) d) m ( m ^ x + 2) <x + m +1
Lai giai
phuong trinh tuang d u o n g vdi ( m - 2)x < 3m - 6
m = 2 bat p h u a n g trinh tro thanh 0 x < 0 s u y ra bat p h u o n g trinh
•Nghiem d u n g vai mgi x.
'^^i m > 2 bat p h u a n g trinh tuong d u o n g vai x < '^"^ = 3 • m - 2
^' m < 2 bat p h u o n g trinh tuang d u o n g vai x > ^"^ ~^ = 3 II luan
m - 2
Phdn lo0i va phuong phdp gidi Dai so'10
m = 2 bat p h u o n g trinh nghiem d u n g voi moi x (c6 tap nghiem la S = m > 2 bat p h u o n g trinh c6 nghiem la x < 3 (c6 tap nghiem la S = (-oo;3) ^ m < 2 bat p h u o n g trinh c6 nghiem la x > 3 (c6 tap nghiem la S = ( 3 ; + G O ) )
b) Bat p h u o n g trinh t u o n g d u o n g voi ( m - 2) x > 4 - m^ ' "
V o l m = 2 bat p h u o n g trinh tro thanh Ox > 0 suy ra bat p h u o n g trinh vo nghiem.
4 - m 2 V o l m > 2 bat p h u o n g trinh tuong d u o n g voi x > m - 2
4- m ^
• = - m - 2
. / . T i l l
V o i m < 2 bat p h u o n g trinh tuong d u o n g voi x < = - m - 2
i q . . . Ke't ludn ' ' . - 4 r ^ ^
m = 2 bat p h u o n g trinh v6 nghiem " " ' ' {') " ' ^
|5;, m > 2 bat p h u o n g trinh c6 nghiem la x > - m - 2 ' • , ' m < 2 bat p h u o n g trinh c6 nghiem la x < - m - 2
c) Bat p h u o n g trinh t u o n g d u o n g voi ( m + 3)^ x > m - 3
V o i m = -3 bat p h u o n g trinh tro thanh Ox > - 6 suy ra bat phuong trinh nghiem d u n g v o i m p i x .
, m - 3 V o i m ^ -3 bat p h u o n g trinh tuong d u o n g voi x > —j
( m + 3) m^&i'
Kel luqn
m = -3 bat p h u o n g trinh nghiem diing v o i moi x . ni - 3 m ?t -3 bat p h u o n g trinh c6 nghiem la x > —2 .
(m + 3)
d) Bat p h u o n g trinh t u o n g d u o n g voi <=> (m^ - l ) x < - 2 m + 1
^ r 1 3
^(m-l)x< V ~ (vim^ + m + l = m + - + - > 0 ) , . „ j
m +m + l y IJ
V o i m = l bat p h u o n g trinh tro thanh Ox<Osuy ra bat p h u o n g trinh ' nghiem.
V o i m > 1 ba't p h u o n g trinh tuong d u o n g v o i x < —
m + m + l . a , : t ) , m - 1
V o i m < 1 bat p h u o n g trinh t u o n g d u o n g v o i x > -
^ ° m + m + l Kel luqn
m = 1 bat p h u o n g trinh v6 nghiem
> 1 bat p h u o n g trinh c6 nghiem la x < ^ m^ + m + 1
m - 1 " ' • "
f(y<l bat p h u o n g trinh c6 nghiem la x > —^ . ' m + m + 1
dti 2. T i m m de bat phuong trinh (m^ - m j x + m < 6x - 2 v6 nghiem.
gat p h u o n g trinh t u o n g d u o n g voi (m^ - m - 6jx < - 2 - m ,!
^ • 2 f m 5^ -2
rang neu m - m - 6 ? t 0 < = > j bat p h u o n g trinh luon c6 nghiem.
Voi m = -2 ba't p h u o n g trinh tro thanh Ox < 0 suy ra bat p h u o n g trinh v6 nghiem
Voi m = 3 bat p h u o n g trinh tro thanh Ox < -5 suy ra bat p h u o n g trinh v6 nghiem
V l y gia trj can t i m la m = -2 va m = 3 . t* iyfi fffi^:(jsF p 2. BAI T A P L U Y $ N T A P M <:> [:
Bai 4.64: Giai va bi^n luan cac bat phuong trinh: ''"^'^ >A
a ) m ( x - m ) < x - l . b) 3x + > m(x + 3). ^ 0
Huong dan gidi ^ a) m(x - m ) < X -1 <=> ( m - l ) x < - 1
N e u : m = l thi 0 x < 2 (diing). Tap nghiem: S=R. . , , N e u : m > l thi x < m + l . Tap nghiem: S= ( - o o; m + 1 ] .
Neu: m < 1 thi x > m + l . Tap nghiem: S=fm + l; + o o ) .
^) 3x + m^ > m(x + 3) o ( m - 3)x < m^ - 3m. 1 ; 1
Neu: m = 3 thi bat p h u o n g trinh Ox < 0: nghiem voi m p i x . rrotjrfcf ^ Neu: m > 3 thi bat p h u o n g trinh c6 nghiem x < m . j ., '^eu: m < 3 thi bat p h u o n g trinh c6 nghiem x > m . .^amy: ,
*'*ô65: a) T i m m de bat p h u o n g trinh mx - 2 < x - m v6 n g h i f m .
' ^'m m de bat p h u o n g trinh m^ (x - 1 ) > 9x + 3 m c6 nghiem d u n g Vx € R . Hu&ng dan gidi * tiu
oat p h u o n g t r i n h t u o n g d u o n g voi (m - 1 ) x < 2 - m ^ '^^ rang neu m^l bat phuong trinh luon c6 nghiem. \
m = 1 bat p h u o n g trinh tro thanh Ox < 1 suy ra ba't p h u o n g trinh
•^ghi^m d i i n g v o l m p i X .
khong CO gia t r i nao ciia m thoa man yeu cau bai toan.
285
b) Ba't phuong trinh tuong duong v6i (m^ - 9 ) x > + 3m
De dang thay ne'u m^ - 9 ;t 0 <=> m ?t +3 thi bat phuong trinh khong nghiem diing Vx e M
Voi m = 3 ba't phuong trinh tro thanh Ox > 18 suy ra bat phuong (- v6 nghiem
Voi m = -3 bat phuong trinh tro thanh Ox > 0 suy ra bat phuong (^j^^, nghiem dung voi moi X . n . j . ^j.-..^
Vay gia trj can tim la m = -3 .