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Nilai x yang memenuhi pertidaksamaan 2log(x2−x−12)<2log(2x+16) adalah ...

Pertanyaan

Nilai x yang memenuhi pertidaksamaan log presuperscript 2 open parentheses x squared minus x minus 12 close parentheses less than log presuperscript 2 open parentheses 2 x plus 16 close parentheses adalah ...

  1. negative 4 less than x less than negative 3 space atau space 4 less than x less than 7

  2. negative 8 less than x less than negative 4 space atau space x greater than 7

  3. negative 4 less than x less than 7

  4. negative 3 less than x less than 4 space atau space 7 less than x less than 8

  5. negative 8 less than x less than 7

R. Hajrianti

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Ingatlah syarat pertidaksamaan logaritma dengan bilangan pokok (basis) a greater than 1, yaitu:

log presuperscript a space f open parentheses x close parentheses less or equal than log presuperscript a space g open parentheses x close parentheses space rightwards arrow space f left parenthesis x right parenthesis less or equal than g open parentheses x close parentheses

dimana f left parenthesis x right parenthesis greater than 0 dan g left parenthesis x right parenthesis greater than 0.

Berdasarkan hal tersebut, maka pertidaksamaan harus memenuhi 3 syarat:

  • log presuperscript a space f open parentheses x close parentheses less or equal than log presuperscript a space g open parentheses x close parentheses space rightwards arrow space f left parenthesis x right parenthesis less or equal than g open parentheses x close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 2 open parentheses x squared minus x minus 12 close parentheses end cell less than cell log presuperscript 2 open parentheses 2 x plus 16 close parentheses end cell row cell x squared minus x minus 12 end cell less than cell 2 x plus 16 end cell row cell x squared minus x minus 2 x minus 12 minus 16 end cell less than 0 row cell x squared minus 3 x minus 28 end cell less than 0 row cell open parentheses x plus 4 close parentheses open parentheses x minus 7 close parentheses end cell less than 0 end table

lakukan uji titik pada garis bilangan sehingga didapat solusi:

H P subscript 1 equals open curly brackets negative 4 less than x less than 7 close curly brackets

  • f left parenthesis x right parenthesis greater than 0

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus x minus 12 end cell greater than 0 row cell open parentheses x plus 3 close parentheses open parentheses x minus 4 close parentheses end cell greater than 0 end table

lakukan uji titik pada garis bilangan sehingga didapat solusi:

H P subscript 2 equals open curly brackets x less than negative 3 space atau space x greater than 4 close curly brackets

  • g left parenthesis x right parenthesis greater than 0

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 16 end cell greater than 0 row cell 2 x end cell greater than cell negative 16 end cell row x greater than cell negative 8 end cell end table

H P subscript 3 equals open curly brackets x greater than negative 8 close curly brackets

Selanjutnya, iriskan solusi ketiga syarat H P subscript 1 equals open curly brackets negative 4 less than x less than 7 close curly brackets, H P subscript 2 equals open curly brackets x less than negative 3 space atau space x greater than 4 close curly brackets, dan H P subscript 3 equals open curly brackets x greater than negative 8 close curly brackets sehingga:

Dengan demikian, nilai x yang memenuhi pertidaksamaan logaritma tersebut adalah negative 4 less than x less than negative 3 space atau space 4 less than x less than 7.

Oleh karena itu, jawaban yang benar adalah A.

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