27,03,2020 Kunlik Mavzu: Kasr–ratsional tengsizliklar

27,03,2020 Kunlik

Mavzu: Kasr–ratsional tengsizliklar.

Agar f(x), (x), g(x), h(x)– ratsional funktsiyalar bo‘lsa, ulardan tuzilgan tengsizliklar quyidagi ko‘rinishga olib kelib yechiladi:

bo‘ladi.

bo‘lsa
1)


bo‘lsa

bo‘ladi.
2)


bo‘lsa

bo‘ladi.
3)


bo‘ladi.
4)


bo‘ladi.
5)


bo‘ladi.
6)
Ushbu tengsizliklar oraliqlar (intervallar) usuli yordamida yechiladi.

Bu usulning mohiyatini quyidagi misollar yordamida tushuntiramiz:

1-misol: (x+1)(x-2)(x-4)>0 ni yeching.

Yechish: (x+1)(x-2)(x-4) ifoda x₁=-1, x₂=2, x₃=-4 bo‘lganda nolga aylanadi. Ushbu sonlarni sonlar o‘qida belgilab, quyidagi (-;-1), (-1;2), (2; 4), (4;+) oraliqlarni xosil qilamiz. (x+1)(x-2)(x-4) ifodaning shu oraliqlardagi ishoralarini aniqlaymiz. Buning uchun ixtiyoriy oraliqqa tegishli sonni berilgan ifodadagi o‘zgaruvchi x o‘rniga qo‘yib, uning ishorasini aniqlash mumkin. Bu masalada (-1; 2) oraliqqa tegishli «0» sonini berilgan ifodaga qo‘yib, ushbu oraliqda uning ishorasi musbat ekanligi aniqlaymiz. Qolgan oraliqlarning ishoralari (-1; 2) oraliqdagi ishoraga mos ravishda aniqlanadi (6-rasm).

6-rasm.
(x+1)(x-2)(x-4)>0 tengsizlik faqat musbat qiymatlarni qabul qilganligi sababli,Echim: (-1;2)(4;+).

2 – misol.2x(x-1)(x+2)(x-3)>0 ni yeching.

Yechish: (7-rasm). x₁=0, x₂=1, x₃=-2, x₄=3

x(-, -2)(0,1)(3, )

7-rasm.

3–misol. ni yeching.

Yechish: (8-rasm).  

8-rasm.
TESTLAR.

1. tengsizlikning manfiy butun yechimlari yig‘indisini toping.

A) –9 B) –12 C) –5 D) –6 E) –4

2. tengsizlikni yeching.

A) B) C) D) (-; -3]  (-2; 1]

E)

3. tengsizlikni yeching.

A) B) C)

D) E)

4. tengsizlikni yeching.

A) ­(1;) B) [1;) C) ) D) ] E) 

5. tengsizlikning butun yechimlari nechta?

A) 4 B) 1 C) 2 D) 3 E) cheksiz ko‘p

6. tengsizlikning butun yechimlari nechta?

A) 4 B) 5 C) 3 D) 2 E) cheksiz ko‘p

7. tengsizlikni yeching.

A) ­(-;-1,5) B) (-1,5;2) C) D) E) ­(-;-2,5)

8. tengsizlikning eng katta va eng kichik butun yechimlari ayirmasini toping.

A) 10 B) 9 C) 8 D) 7 E) 11

9. tengsizlikning yeching.

A) ­(-;-1) (0;1) B) [0;1) C) D)  E) ­(-;-1)

10. tengsizlikning eng katta va eng kichik butun yechimlari ayirmasini toping.

A) 6 B) 4 C) 5 D) 2 E) 3

11. tengsizlikni yeching.

A) (2; ) B) (-; 2] C) (-; 2) D) [2; ) E) 

12. tengsizlikning butun sonlardan iborat yechimlari yig‘indisini toping.

A) 12 B) 8 C) 10 D) 15 E) 0

13. tengsizlikni yeching.

A) (-; 1)[2; ) B) [1; 2] C) [1; 2) D) (-1; 2] E) (-; 1]

14. (x-2)(x-3)<0 tengsizlikni yeching.

A) (-3; 2) B) (-; -2)  (3; ) C) (-; -3)  (2; )

D) (-;2)  (3; ) E) (2; 3)

15. tengsizlikni yeching.

A) (-2; 1) B) (1; 2) C) [1; 2)

D) (-; 1)  (2; ) E) (-; 1]  (2; )

16. Nechta tub son tengsizlikning yechimi bo‘ladi?

A) 3 B) 4 C) 2 D) 1 E) 5

17. Nechta tub son tengsizlikni yechimi bo‘ladi?

A) 7 B) 2 C) 3 D) 5 E) 11

18. Nechta tub son tengsizlikning yechimi bo‘ladi?

A) 7 B) 5 C) 1 D) 2 E) 3

19. tengsizlikning barcha butun yechimlari yig‘indisini toping.

A) 1 B) 4 C) 5 D) 2 E) 3

20. tengsizlikning barcha butun yechimlari yig‘indisini toping.

A) 6 B) 7 C) 8 D) 9 E) 11

21. tengsizlik o‘rinli bo‘ladigan n ning barcha natural qiymatlari yig‘indisini toping.

A) 4 B) 2 C) 5 D) 3 E) 6

22. x⁵-16x>0 tengsizlikning eng kichik butun musbat va eng katta butun manfiy yechimlari ko‘paytmasini toping.

A) - 5 B) - 4 C) - 6 D) - 2 E) –3

23. tengsizlikni yeching.

A) ­(- ; - 3) B) (- 3; - 3) C) D)  E) ­(- ; )