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![](/i/favi32.png) F u s m o n o V, R. I s o m o V, B. X o ‘ j a y e V matematikadano o ).
3 - m i s о 1. 9 - 7 x > - l - 1 7 x
tengsizlikni yeching.
Y ech ilish i: 9 - 7x > -1 - 17x
17x -
l xUsmanov F. Matematikadan qo\'llanmao o ).
3 - m i s о 1. 9 - 7 x > - l - 1 7 x
tengsizlikni yeching.
Y ech ilish i: 9 - 7x > -1 - 17x<=>
<£=> 17x -
l x
> -1 - 9 <=> lOx > - 10 => [x > - 1.
Tengsizlikni q a n o a tla n tiru v c h i so n la r to 'p la m i 17-rasm da ta s
v irla n g a n .
J a v о b: x e (-1 ; +
o o ).
4-m i s о 1. -
> 0 tengsizlikni yeching.
0
3
x
15-rasnt
--- .--
0
1
X
16-rasm
д / / / у / / / / / / / / / / / / / / / У ^
-1
О
X
17-rasm
80
Y e c h i l i s h i :
3-л -
> 0 <=>
<=>---- г > 0 <=> .v - 3 > О =>
[х >
3.
х - з
1
T engsizlikni q a n o a tla n tiru v c h i
so n la r to 'p la m i 18-rasm da tasvir-
lan g an .
J a v o b : x e ( 3 ; +oo).
18-rasm
5-m i s о 1.
yeching.
2 + x
Y e c h i l i s h i :
< 0 tengsizlikni
< 0 <=>
-2
0
19-rasm
LULLLLLlLtLL
20-rasm
2 + x
<=> 2 + x
< 0 =>
[x
> -2 .
T engsizlikni q a n o a tla n tiru v c h i
s o n la r 1 9 -ra s m d a ta s v ir la n g a n .
J a v о b:
x
e (-oo; -2 ).
2
x
3
x
—
1
6-m i s о 1. л :
-— <
——
tengsizlikni yeching.
v
, . . . , •
2x + 3
x - 1
1 2 x - 8 x - 12- З х + З
л
Y e c h i l i s h i : д :-------— <
——
о ------------
—
------------ < 0<=>
< = > x - 9 < 0 = > [ x < 9.
T engsizlikni q a n o a tla n tiru v c h i so n la r to 'p la m i 20-rasm da ta s
v irla n g a n .
J a v о b:
x <
9.
10-§. C h iziq li te n g sizlik la r siste m a si
T a ’ r i f .
Chiziqli tengsizliklar sistem asi deb, bir x il noma lum
о ‘zgaruvchiga ega bo ‘Igan ikki y o k i undan ortiq chiziqli tengsizliklar
to ‘plam iga aytiladi.
U sh b u sistem alar chiziqli tengsizliklar sistem alariga m isol b o 'la
oladi:
1)
5.x + 6 < x,
3x + 12 < x + 17.
2
)
Г3,3 - 3(1,2 - 5 x ) > 0 ,6 (1 0 x + 1),
3 ) | l , 6 - 4 , 5 ( 4 x - l ) < 2 x + 2 6 ,l.
4)
2 (x - 1) - 3 (x - 2) < x,
6 x - 3 < 17 - ( x - 5).
2 x - 1 < x + 3,
5x - 1 > 6 - 2x,
x - 5 < 0.
81
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-- •
►
- 2 - 1 5 - 1
О
1
2 2 ,5 3
21-rasm
-
2
-
1
0
1
2
3 З у
22-rasm
- 1
0 0, 1
I
а
23-rasm
10.1.
C hiziqli te n g siz lik la r
sistemasini yechish.
Tengsizliklar
sistem asin i yechish noma ’lum
о ‘zgaruvchining sistemaning har
q a y s i
t e ng s i z l i g i ni
q a n o a t l a n t i r a d i g a n
b a rch a
q iy m a tla r i to ‘p la m in i to p ish
dem akdir.
Y u q o r i d a
k e l t i r i l g a n
sistem alarni yecham iz:
\5 x
+
6 <
a
,
1)1зд:
+ 12 <
a
+ 1 7 . ^
4 x
< - 6 ,
0
| 3 x - x < 1 7 - l 2 W | 2 . v < 5
^
5 a
- x <,
-
6
,
3 a - A <
x <
-1 ,5 ,
a
< 2 , 5
= ф [л' ^ - 1’5 -
S istem aning yechim lari birlashm asi 21-rasm d a tasvirlangan:
J a v о b:
a
e (^ o ; -1,5].
Г2(.v - 1) - 3 ( a - 2 ) <
a
,
[ 2 a - 2 - 3 a +
6
-
a
< 0 ,
2
) [ 6 a - 3 < 17 - ( a - 5 ) .
° [ 6 a - 3 + a < 1 7 + 5
^
- 2
a
< - 4 .
7
a
< 25
:
2 <
а
<
з
;
S istem aning yechim lar kesishm asi 22-rasm da tasvirlangan:
J a v о b:
a
e 2;3
4
.
3 , 3 - 3 ( 1 , 2 - 5 a ) > 0 , 6 ( 1 0 a + 1),
j
3,3 - 3 ,6
+
1 5 a
> 6
a
+
0,6,
1 , 6 - 4 , 5 (4
a
- l ) < 2
a
+ 26,1
9
a
> 0 ,9 ,
Г а
> 0 ,1 ,
3)
^ [1,6 - 18
a
+ 4,5 < 2
a
+ 26,1 ^
[
a
< 0 , 1 .
S istem aning yechim lar birlashm asi 23-rasm da tasvirlangan:
J a v o b : (0,1; +oo).
82
2,v-l 2 x - x < 3 + 1,
■
fx < 4,
4)
^ - \> 6 - 2 x , ^
■5jv +
2
.x
> 6 + 1, . *l x < 5 , 0
v-5<0
x < 5.
l x >
7.
1-
f x < 4 ,
U > i :
=> [1 < х < 4 .
S istem aning yechim lari kesishm asi 24-rasm da tasvirlangan:
J a v о b:
x g
[1; 4).
5) U shbu
i v _l
I < ~ — < 2
qo'sh tengsizlikni
yeching.
Y e c h ilis h i. B erilg an q o 's h ten g sizlik q u y id a g i te n g siz lik la r
sistem asiga teng kuehlidir:
2 x
- 1
2
2 x - \
> 1.
<2
Bu sistem ani yecham iz:
2 x - l
2 x
- 1 > 2,
2
2.V-1
> 1,
2 x > 3,
< 2 <* ) 2
x
- \
< 4 ^ Ь х < 5 ^
x > 1,5,
x < 2,5 "
6
)
Y e c h i l i s h i :
[1,5 < x < 2,5.
J a v o b : x
g
(1,5; 2,5).
f 14x - 3 < 7 + 9 x ,
11 < x - 3
[14x - 3 < 7 + 9 x ,
[l < x - 3
K o o r d i n a t a
t o ' g ' r i
ch izig 'id a sistem aning h a r bir
te n g s iz lig i
y e c h im in in g
tasv irid a n k o 'rin ib tu rib d ik i
( 2 5 - r a s m ) , x < 2 v a x > 4
t e n g s i z l i k l a r n i
q a n o a t l a n t i r u v c h i s o n l a r
to 'p la m i um um iy elem entga
e g a e m a s , y a ’n i u l a r n i n g
kesishm asi b o 's h to 'p la m .
tengsizliklar sistem asini yeching
5x < 10,
<=>
x > 4
x < 2,
x > 4 .
0
I
24-rasm
0
25-rasm
83
H a q iq a ta n ham , 2 d an k a tta em as va 4 d a n k a tta b o ‘lishi k erak
b o ‘lgan son m av ju d em as.
Jav o b :
x
e 0 (b u n d a 0 - b o ‘sh t o ‘p lam belgisi).
U - § . C h iziq li te n g sizlik la rn i yech ish n in g
g ra fik usuli
11.1.
kx > n
tengsizlikni yechishning grafik usuli.
K o o rd in a ta la r
tek islig id a
у
=
kx
va
у = n
fu n k siy a la rn in g g rafik la rin i yasaym iz. A g a r
к *
0 b o ‘lsa,
у
=
k x
to 'g 'r i chiziq a lb a tta
у
=
n
to 'g 'r i chiziqni kesadi
(k >
0, 26-rasm ;
к
< 0, 27-rasm ). Bu to 'g 'r i chiziqlarning kesishish nuqtasini
M
harfi
bilan belgilaymiz. Bu n u qtaning abssissasifoc = и tenglam aning ildizi
x =
j
ga teng. A gar
к >
0 b o 'lsa, 26-rasm da k o 'rin ib turg an id ek ,
barcha
x > j
lar uchun
kx > n
tengsizlik to 'g 'r i tengsizlik b o 'lad i.
A gar A- < 0 b o 'lsa (27-rasm ), barcha
x < j
lar uchun
k x > n
b o 'lad i.
11.2. Bir o'zgaruvchili chiziqli tengsizliklar sistemasini grafik
usul bilan yechish.
U sh b u m asalani qaraylik: « O 'zg aru v ch i .v ning
q an d a y q iy m atlarid a
у - x
+ I va
у = - 2 x + 4
fu n k siy alarn in g h a r
ikkisi ham n o m an fiy q iy m atlar q ab u l qiladi?»
84
Berilgan funksiyalarning grafiklarini bir chizm ada tasvirlash yo'li
bilan bu m asalani osongina yechish m um kin. Bu grafiklar 28-rasm da
ta sv irlan g an .
O 'zg a ru v ch i x ning iz lan ay o tg an q iy m atla ri to 'p la m i [-1; 2] kes-
m ad an ib o ra t ekan.
12-§. N o m a ’lu m la ri m o d u l b e lg isi o stid a bo'lga n
te n g la m a la r va te n g s izlik la r
12.1.
M odulli ten g lam ala r. B a ’zi te n g la m a la rn in g n o m a ’lu m
lari m o d u l (a b so lu t q iy m at) belgisi o stid a b o 'la d i. B u n d ay te n g
lam alarn in g yechilishi b iro r o 'z g a ru v c h i x m iq d o rn in g a b so lu t q iy
m ati
a
m u sb a t so n g a ten g b o 'ls a , b u h o ld a x nin g o 'z i yo
a
ga, yoki
- a
ga ten g b o 'lis h ig a a so sla n a d i. M a sa la n , |x| = 5 b o 'ls a , x = 5
yoki x = - 5 .
Bir n echa m iso llar qaraylik:
1-m i s о 1. |x - l| = 2 tenglam ani yeching.
Y e c h i l i s h i : t a ’rifga k o 'r a
J x - 1 > 0,
J x > 1,
j x - 1 = 2 ^ j x = 3
O 'zg aru v ch i x ning bu ikkala qiym ati ham berilgan tenglam ani
q a n o a tla n tira d i.
J a v o b : 3; -1 .
2-m i s о 1.16 — 2 x | = 3x + 1 ten g lam an i yeching.
Y e c h i l i s h i : t a ’rifga k o 'r a
f 6 - 2 x > 0 ,
f - 2 x > - 6,
H a q iq a ta n ham , x = - 7 berilgan tenglam ani q anoatlantirm aydi:
x = - 7 d a |6 - 2 x | = |20| = 20, 3x + 1 = -2 1 + 1 = - 2 0 ,2 0 * -2 0 .
J a v o b : 1.
yoki
{:
6 - 2 x = 3x + l ^ j - 5 x = - 5
^
6 - 2 x < 0,
J - 2 x < - 6,
6 - 2 x = - 3 x — 1 => 1 x = —7
85
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