Irratsional tenglama va tengsizliklarni yechish. Reja

Aim.uz

Irratsional tenglama va tengsizliklarni yechish.
Reja :

1. 
   Irratsional tenglamalarni yechish usullari
   
2. 
   Irratsional tengsizliklarni yechish usullari
   

Noma’lum qatnashgan ifoda ildiz belgisi ostida qatnashgan tenglamalar irratsional tenglamalat deyiladi :

Irratsional tenglamalar xususuiy hollarda quyidagi ko’rinishlarda bo’lishi mumkin.

1. 
   Bitta kvadrat ildiz qatnashganirratsional tenglama
   

Misol : tenglamani yeching.
Tenglamaning aniqlanish sohasi (q.q.q.s).
;

2. 
   Ikkita kvadrat ildiz qatnashgan tenglama.
   

Misol: tenglamani yeching.
;

3. 
   Bu xil tenglamalarni sun’iy usullar bilan ham yechish mumkin
   

Misol: tenglamani yeching.
Tenglamaning aniqlanish sohasini, ya’ni D(T) ni topamiz: q.q.q s yoki
almashtirish bajarilsa, u holda bo’lib, tenglama hosil bo’ladi, buni yechilsa, ekanligi kelib chiqadi.
Misol : tengsizlikni yeching.
q.q.q.s .
Chet ildiz hosil bo’lganligi uchun bunday hulosa yuritamiz :

1. 
   x<1 da chap tomon manfiy; o’ng tomon manfiy emas.
   

Demak,

2. 
   x ≥1 da chap va o’ng tomonlar musbat.
   

4x²-8x+4<4x²+23x+15;
. Bu holda yechim yo’q.
2. Irratsional tengsizlik va ularni yechish.
1-misol: 2(x-1)< tengsizlikni yeching.
Yechish:
Chet ildiz hosil bo’lmasligi uchun bunday mulohaza yuritamiz:
a) x<1 da chap tomon manfiy; o’ng tomon manfiy emas. Demak,
b) x 1 da chap va o’ng tomonlar musbat. 4x²-8x+4<4x²+23x+15; . Bu holda yechim yo’q.
Javob:

2-misol: x³+x²+2 >4 tengsizlikni yeching.

Yechish: f(x)= x³+x²+2 -4 funksiya [0; ) oraliqda o’suvchi va aniqlangan bo’lib, f(1)=0 bo’lganidan x>1 bo’ladi. Demak, yechim (1; ) oraliqdan iborat.
3-misol. (x-4) tengsizlikni yeching.
Yechish: Bu tengsizlikni yechish unga teng kuchli bo’lgan

sistemani yechish bilan bog’liq.
Demak, yechim {(- ;4] (- ;-2] [1; )}=(- ;-2] [1;4]
Yuqoridagilardan ba’zan irratsional tengsizliklarni yechish tengsizliklar sistemasini yechish bilan ekvivalent bo’lishi mumkinligi ko’rinadi.


Mustaqil yechish uchun misollar:

1. 
   5+ >7;
   
2. 
   x+ <8;
   
3. 
   >3;
   
4. 
   <4;
   
5. 
   >1.
   

Tayanch iboralar:
irratsional, tenglama, tengsizlik, musbat, manfiy, chet ildiz


Nazorat savollari:

Tenglamalarni yeching:

1.
2.
3.
4.

Tengsizlikni yeching

1.

2.
3.
Test savollari

1. tenglamani yeching.

A) 6
B) 5

C) -9

2. tenglamani yeching.

1. 
   -2;5
   
2. 
   2;5
   
3. 
   -5;2
   

3. tengsizlikni yaching

A)

B)
C)
4. tenglamani yeching.
A) – 1 B) – 2 C) 2 D) 0
E) to’g’ri javob berilmagan
5. Agar bo’lsa, х + у ni toping.
A) 2 B) 3 C) 4 D) 5 E) 6
6. tenglik x ning qanday qiymatlarida o’rinli
A) – 1 B) 1 C) – 3
D) hech qanday qiymatida E) 3
7. Agar bo’lsa, ni toping
A) 17 B) 18 C) 19 D) 16 E) 15
8. tenglamalar sistemasini yeching.
A) B) C)
D) E)
9. Agar tenglamaning katta ildizi x₀ bo’lsa, x₀ + 10 nechaga teng?
A) 10 B) 12 C) 20 D) 15 E) 18
8. tenglamaning ildizlari yig’indisini toping.
A) 0 B) –2 C) –4 D) 2 E) 4
9. tenglamaning ildizlari yig’indisini toping.
A) 4 B) –3 C) 3 D) –4 E) –5
10. Agar bo’lsa,
ning qiymatini toping.
A) 18 B) 20 C) D)
E)
11. tenglamaning natural ildizlari nechta?
A) B) 1 C) 2 D) 3 E) 4
12.
tenglamaning ildizlari yig’indisini toping.
A) B) 4 C) 2 D) 8 E) 9
13. va bo’lsa,
nimaga teng?
A) 6 B) 4 C) 5 D) 3 E) 8
14. tengsizlikning yechimini ko’rsating
A) [3; ∞) B) [1; 2]
C) [3; 1] [2; ∞) D) [2; ∞)
E) (∞; 2] [1; ∞)
15. Quyidagilardan qaysi biri
tengsizlikning yechimi?
A) (–∞; 3] B) (–∞; –2] U [1; 3]
C) [–2; 3] D) [–1; 2] U [3; ∞) E) [–2; ∞)

Aim.uz