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Lineêre Ongelykhede

Ondersoek : Ongelykhede op 'n Getallelyn

Stel die volgende voor op getallelyne:

  1. x = 4 x = 4
  2. x < 4 x < 4
  3. x 4 x 4
  4. x 4 x 4
  5. x > 4 x > 4

'n Lineêre ongelykheid is soortgelyk aan 'n lineêre vergelyking aangesien die hoogste eksponent van die veranderlike 1 is. Die volgende is voorbeelde van lineêre ongelykhede:

2 x + 2 1 2 - x 3 x + 1 2 4 3 x - 6 < 7 x + 2 2 x + 2 1 2 - x 3 x + 1 2 4 3 x - 6 < 7 x + 2
(1)

Die metodes wat gebruik word om lineêre ongelykhede op te los, is identies aan dié wat gebruik word om lineêre vergelykings op te los. Die enigste verskil kom voor wanneer daar 'n vermenigvuldiging met of deling deur 'n negatiewe getal is. Ons weet byvoorbeeld dat 8>68>6. As beide kante van die ongelykheid gedeel word (byvoorbeeld deur -2-2), sien ons -4-4 is nie groter as -3-3 nie. Dus moet die ongelykheid omkeer, wat beteken -4<-3-4<-3.

Tip:

Wanneer beide kante van 'n ongelykheid met 'n negatiewe getal vermenigvuldig word, of met 'n negatiewe getal gedeel word, verander die rigting van die ongelykheid. Om hierdie rede mag ons nie met 'n veranderlike vermenigvuldig as ons nie weet nie wat die onbekende (veranderlike) se teken is nie.

Byvoorbeeld, as x<1x<1, dan -x>-1-x>-1.

Om 'n ongelykheid met 'n gewone vergelyking te vergelyk, sal ons eers die gewone vergelyking oplos. Los op 2x+2=12x+2=1.

2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2 2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2
(2)

As ons hierdie antwoord op 'n getallelyn voorstel, kry ons:

Figure 1
Figure 1 (MG10C10_001.png)

Kom ons los nou die ongelykheid 2x+212x+21 op:

2 x + 2 1 2 x 1 - 2 2 x - 1 x - 1 2 2 x + 2 1 2 x 1 - 2 2 x - 1 x - 1 2
(3)

As ons hierdie antwoord op 'n getallelyn voorstel, kry ons:

Figure 2
Figure 2 (MG10C10_002.png)

Soos jy kan sien, vir die vergelyking is daar slegs 'n enkele waarde van xx waarvoor die vergelyking waar is. Vir die ongelykheid is daar egter 'n hele versameling waardes waarvoor die ongelykheid waar is. Dit is die groot verskil tussen gewone vergelykings (gelykhede) en ongelykhede.

Figure 3
Khan Akademie video oor ongelykhede - 1

Figure 4
Khan Akademie video oor ongelykhede - 2

Exercise 1: Lineêre Ongelykhede

Los op vir rr: 6-r>26-r>2

Solution

  1. Step 1. Kry al die konstantes aan die RK :
    - r > 2 - 6 - r > - 4 - r > 2 - 6 - r > - 4
    (4)
  2. Step 2. Vermenigvuldig weerskante met -1 :

    Wanneer jy met 'n negatiewe getal vermenigvuldig, draai die rigting van die ongelykheid om.

    r < 4 r < 4
    (5)
  3. Step 3. Stel die antwoorde grafies voor :

    Figure 5
    Figure 5 (MG10C10_003.png)

Exercise 2: Lineêre Ongelykhede

Los op vir qq: 4q+3<2(q+3)4q+3<2(q+3) en stel die oplossing voor op 'n getallelyn.

Solution

  1. Step 1. Brei alle hakies uit :
    4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6 4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6
    (6)
  2. Step 2. Kra al die konstantes aan die RK en al die onbekendes aan die LK :
    4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3 4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3
    (7)
  3. Step 3. Los die ongelykheid op :
    2 q < 3 deel beide kante deur 2 q < 3 2 2 q < 3 deel beide kante deur 2 q < 3 2
    (8)
  4. Step 4. Stel die oplossing grafies voor :

    Figure 6
    Figure 6 (MG10C10_004.png)

Exercise 3: Saamgestelde Lineêre Ongelykhede

Los op vir xx: 5x+3<85x+3<8 en stel die antwoord op 'n getallelyn voor.

Solution

  1. Step 1. Trek 3 af van die linkerkant, middel en regterkant van die ongelykhede :
    5 - 3 x + 3 - 3 < 8 - 3 2 x < 5 5 - 3 x + 3 - 3 < 8 - 3 2 x < 5
    (9)
  2. Step 2. Stel die antwoord grafies voor :

    Figure 7
    Figure 7 (MG10C10_005.png)

Lineêre Ongelykhede

  1. Los op vir xx en stel die oplossing grafies voor:
    1. 3x+4>5x+83x+4>5x+8
    2. 3(x-1)-26x+43(x-1)-26x+4
    3. x-73>2x-32x-73>2x-32
    4. -4(x-1)<x+2-4(x-1)<x+2
    5. 12x+13(x-1)56x-1312x+13(x-1)56x-13
     
    Kliek hier vir die oplossing
  2. Los die volgende ongelykhede op. Illustreer jou antwoord op 'n getallelyn as xx 'n reële getal is.
    1. -2x-1<3-2x-1<3
    2. -5<2x-37-5<2x-37
    Kliek hier vir die oplossing
  3. Los op vir xx: 7(3x+2)-5(2x-3)>77(3x+2)-5(2x-3)>7
     
    Illustreer die antwoord op 'n getallelyn.
     
    Kliek hier vir die oplossing

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