Nejedn Rjesenja 6 a Za Web

8/2/2019 Nejedn Rjesenja 6 a Za Web

1/22

**** MLADEN SRAGA ****2010.

UNIVERZALNA ZBIRKA

POTPUNO RIJEENIH ZADATAKA

PRIRUNIK ZA SAMOSTALNO UENJE

1

UREAJU NA SKUPU REALNIHBROJEVA

JEDNADBENEJEDNADBE

APSOLUTNE JEDNADBE

APSOLUTNE NEJEDNADBE


2/22

Rjeenja Ureaj na skupu realnih brojeva

Autor:

MLADEN SRAGA

Grafiki urednik:

Mladen Sraga

BESPLATNA - WEB-VARIJANTA

Dodatne upute i VIDEO objanjenja uz sve zadatke iz ove web-varijante biti e

objavljene na ovoj naojweb-stranici: http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htmdovoljno je da dvokliknete na ovaj plavi dio teksta i stranica bi se trebala otvoriti

Tisak:M.I.M.-SRAGA d.o.o.

CIP-Katalogizacija u publikaciji Nacionalna i sveuilina knjinica, Zagreb

M.I.M-Sraga d.o.o. 2010.

Potpunu garanciju na kompletnu zbirku daje: centar za dopisnu poduku M.I.M.-SRAGA -

dakle sve to vam se ini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-123i traite dodatne upute i objanjenja ...Dodatne upute i objanjenja moete zatraiti i na mail: [email protected]

M.I.M.-SRAGA d.o.o. zadrava sva prava na reproduciranje , umnaanje , prodaju ovezbirke potpuno rijeenih zadataka iskljuivo u okviru svog programa poduke i dopisnepoduke.Nikakva komercijalna upotreba ove zbirke nije dozvoljena bez pismene dozvole nakladnika!

2autor: Mladen Sraga
http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htmmailto:[email protected]:[email protected]://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm


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Linearne JEDNADBE

( ) ( ) ( ) ( )

( )

2 2 2 2 2 2

2 2 2

2 2 2

2 2

kvadriramo po pravilima4) 1 2 2 1 2

2 1 4 4 2 2 1

2 4 1 4 2 4 2

2 2 2 4 2 1 4

6 3 : 6

3

61

2

301.x x x a b a a

x x x x x x

x x x x x x

x x x x

x

x

x

+ + = = +

+ + + + = +

+ + + + = +

+ + =

=

=

=

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak

http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm

ili

direktni link na : video uputu uz 301. 4)

b b

( )( )

( )( ) ( ) ( ) ( )( )

( )2

3 114) 2 3 3 uvjet: 3 0 , 3 0

3 33 3

3 3 1 3 2 3 3

3

x xx x x x

x xx x

x x x x x x

x

++ = + +

+

+ + + = +

+ ( )

( )

2 2

2 2 2

2 2 2

3 3 2 9

6 9 4 3 2 18

2 6 4 9 3 18

2 30 : 2

15

x x x x

x x x x x

x x x x x

x

x

+ + + =

+ + + + =

+ + =

=

=

Ovo nisu svi zadaci iz ove zbirke ,Ovo je samo manji dio od oko 10% zadataka iz kompletne zbirke i ovdje su postavljeni samo kao ogledni primjerci .Ali vam mogu posluiti kao solidna vjeba pred testove ili ispitivanja u koli

3www.mim-sraga.com

M.I.M.-Sraga centar za poduku
http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htmhttp://www.youtube.com/watch?v=TgWv6CUaKLE&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.youtube.com/watch?v=TgWv6CUaKLE&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm


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1136.

NEJEDNADBE

Tehnika rjeavanja linearnih nejednadbi slina je rjeavanju linearnih jednadbi nepoznanice idu lijevo abrojevi desno. Kada nejednadbu mnoimo ili dijelimo negativnim brojem mijenjamo joj smisao:

Primjer: ( )

dakle samo okrenemo znak nejednakosti !

2 4 / : 2

2

x

x

<

> Nekoliko primjera:

+

Pogledaj dodatnu VIDEO uputu

uz ovaj zadatak

direktni link

na uputu: zadatak: 1136. 1) do 4)
http://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4fin


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1136.

( )

1 1 2 15) 6

2 3 3 2

3 2 2 2 3

3 2 4 3

3 4 3 2

1 / : 1

1

x x

x x

x x

x x

x

x

+ < +

+ < +

+ < +

<

<

>

Pogledaj dodatnu VIDEO uputu

uz ovaj zadatak na YouTubedirektni link

na uputu: zadatak: 1136. 1) do 5)

+11,x +

( ) ( ) ( )

( )

1 1 26) 6

2 3 2

3 1 2 1 3 2

3 3 2 2 3 6

3 2 3 6 3 2

2 11 / : 2

11

2

x x x

x x x

x x x

x x x

x

x

+

+

+ +

jo jedan zadatk na tu temu:

6.a

11 +12

11,

12x

+

5www.mim-sraga.com

http://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=IZD_1wa_nEg&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcphttp://www.youtube.com/watch?v=IZD_1wa_nEg&list=UU_OImhcINU0KEQhZ-64S-eQ&index=1&feature=plcp


6/22


1137. Sustav linearnih nejednadbi

U ovom zadatku rjeavamo sustav dvije linearne nejednadbe. Tehnika rjeavanja je takva da svakunejednadbu rijeimo za sebe isto kao to smo to radili u zadatku 1.136Rjeenje sustava je presjek rjeenja prve i druge nejednadbe. Dakle na istom brojevnom pravcu nacrtamorjeenja obadvije nejednadbe tako da svako rjeenje ima svoj nagib ( rafuru )

rjeenje ove jed. je: rjeenje ove jed. je:

pokaimo to na 1. zadatku:

prvo rjeimo svaku nejednadbu za sebe:

1) 3 0 , 5 0

3 5

x x

x x

+ +

+ > +

>

>

5)

10 47

10 , 47

x

YouTube direktni link

na uputu uz: zadatak: 1137. 5) do 6)

( ) ( ) ( ) ( )

( )

3 1 1 4 3 11 6 , 2 62 3 2 3

3 3 2 1 6 3 1 4 2 3 1 12

3 9 2 2 6 3 12 6 2 12

3 2 2 6 9 12 6 2 12 3

5 18 11 / : 18

11

18

x x x x

x x x x

x x x x

x x x x

x x

x

+ + + > +

+ + > + +

+ + > + +

+ > +

>

<

6)

11

185

, 5x

http://www.youtube.com/watch?v=OwROeT1Mte4&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=OwROeT1Mte4&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4fin


8/22


( ) ( )2) 2 3 0

I I

2 0 , 3 0 2 0 , 3 0

2 3 2

x x

x x x x

x x x x

+ + < < + >

> <

I

1


na uputu uz: zadatak: 1138.a. pod - 1)

, 1x

[, 1 1,x +

+11

ukupno rjeenje je: I. II.

( )( )

1138.

a)

1) 1 1 0I I

1 0 , 1 0 1 0 , 1 0

1 1 1

x x

x x x x

x x x x

+

+ +

1 1 + 1 1

[1,x +

3 2 3 2x 3,2x


3,2x

http://www.youtube.com/watch?v=EDPpmRxUgE4&feature=plcp&context=C3e81c39UDOEgsToPDskKNKaxWkoYnxCeBnP6NUufjhttp://www.youtube.com/watch?v=EDPpmRxUgE4&feature=plcp&context=C3e81c39UDOEgsToPDskKNKaxWkoYnxCeBnP6NUufjhttp://www.youtube.com/watch?v=EDPpmRxUgE4&feature=plcp&context=C3e81c39UDOEgsToPDskKNKaxWkoYnxCeBnP6NUufj


9/22


28) 0

2

I I

2 0 , 2 0 2 0 , 2 0

2 2 2

x

x

x x x x

x x x x

>

+

> + > < + >

tono

svi

su rj.

x R

netono

pa I otpada

I

3 +

3,x +


3,x +

YouTube direktni linkna uputu uz : zadatak: 1138.b. pod - 7)

http://www.youtube.com/watch?v=vU1xobyBVGI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=vU1xobyBVGI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=vU1xobyBVGI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4fin


11/22


( )

( ) ( )

" jedinicu prebaci na ljevu stranu "

1138.

c)

11) 1

1

11 0

1

1 1 10

1

1 10

1

20

1

I II

2 0 , 1 0 2 0 , 1

2 / : 1 1 2 / : 1 1

2 2

x

x

x

x

x

x

x

x

x x x

x x x

x x

>

>

>

+>

>

> > < > <


na uputu uz: zadatak: 1138.c. pod - 1)

0x

x

1 221

x 1,2x


1,2x

( ) ( )( ) ( )

1 41138. ) 18) 0

1 2

x xd

x x

+

video uputa i objanjenje uz ovaj: zadatak: 1138.d) pod - 18) na YouTube

11www.mim-sraga.com

http://www.youtube.com/watch?v=xwK4f8FoRZI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=xwK4f8FoRZI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4finhttp://www.youtube.com/watch?v=zrBpa2MZ8Nkhttp://www.youtube.com/watch?v=zrBpa2MZ8Nkhttp://www.youtube.com/watch?v=zrBpa2MZ8Nkhttp://www.youtube.com/watch?v=xwK4f8FoRZI&feature=plcp&context=C30fac59UDOEgsToPDskLn0YwTGa_LotVhCASg4fin


12/22


1138. d)

( )( )1 43) 0

3

1 0 4 0 3 0

1 4

x x

x

x x x

x x x

+

= + = == =

uvjet:

3 0

3

x

x

3=

4 1 3

4 0

1 0

03

0

x

x

x

+

+ + + +

+ +

+

+ +

[ ]4,1 3,x +

( )( )

( )

2 34) 0

1

2 0 3 0 1 0

2 3 / : 1

3

x x

x

x x

x x

x

+

+

+ = = + =

= = =

=

uvjet:

1 0

1

x

x

+

1

x

x

2 1 3

2 0

1 03 0

0

x

x

x

+

+ + + +

+ + + + + +

+ +

, 2 1,3x

( )( )

( )( )

1 41138. ) 18) 0

1 2

x xd

x x

+

video uputa i objanjenje uz ovaj:

zadatak: 1138.d) pod - 18) na YouTube

http://www.youtube.com/watch?v=zrBpa2MZ8Nkhttp://www.youtube.com/watch?v=zrBpa2MZ8Nkhttp://www.youtube.com/watch?v=zrBpa2MZ8Nk


13/22


13www.mim-sraga.com

1170.

( ) ( )

( )

( )

( ) ( )( )( )

( ) ( )

( ) ( )

( )( )

( )

2 2

2 2

Izraunaj

) 2 1 2 3 2 1 2 3

2 1 2 3

2 1 2 3

2 1 2 3

2 1 2 3 2 2 1 3

2 2 4

3 2 3 2 2 3 2 3 2 2

3 2 3 2 2

3 2 3 2 2

3 2 3 2 2

3 2 3 2 2 3 3 2 2 2

2 3 3 2

+

+

= =

= =

= +

= + =

= + = +

=

=

= =

=

= + =

= + = + =

=

1

2)

Pogledaj dodatnu VIDEO uputu uz ovaj zadatak

http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm

ili

direktni link na YouTube-

http://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htmhttp://mim-sraga.com/Zbirka-potpuno-rijesenih-zad-Mat-1-uredjaj.htm


14/22


1184.

1) 5x =

5x = 5x = +

2) 1 2x =

1 2

1 21

x

xx

=

= =

1 2

1 23

x

xx

= +

= +=

3) 2 5 3x =

2 5

2 5 3

2 8 / : 2

4

x

x

x

x

3 = += +

=

=

2 5

2 5 3

2 2 / :

1

3

2

x

x

x

x

=

=

=

=

YouTube direktni link na uputu uz: zadatke: 1184. od - 1) do 4)

4) 3 4 2x =

( )

3 4 2

4 3 2

4 5 / : 4

5

4

x

x

x

x

=

=

=

=

( )

3 4 2

4 2 3

4 1 / : 4

1

4

x

x

x

x

= +

=

=

=

http://www.youtube.com/watch?v=bWUwjkj19Yw&feature=plcp&context=C36f1fd9UDOEgsToPDskJbH8OaaS1rNs1w_2lF-0Qlhttp://www.youtube.com/watch?v=bWUwjkj19Yw&feature=plcp&context=C36f1fd9UDOEgsToPDskJbH8OaaS1rNs1w_2lF-0Qlhttp://www.youtube.com/watch?v=bWUwjkj19Yw&feature=plcp&context=C36f1fd9UDOEgsToPDskJbH8OaaS1rNs1w_2lF-0Ql


15/22


1185.3) 2 5 3 1x x = +

( )

( )

za: 2 5 0

2 5 / : 2

5

2

2 5 3

2 5 3 1

2 3 1 5

6 / : 1

6

x

x

x

1x x

x x

x x

x

x

+ = +

= +

= +

=

=

( )

( )

za: 2 5 0

2 5 / : 2

5

2

2 5 3 1

2 5 3 1

2 3 1 5

5 4 / :

4

5

x

x

x

5

x x

x x

x x

x

x

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Nejedn Rjesenja 6 a Za Web