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Penyelesaian dari begin mathsize 14px style fraction numerator 1 over denominator x minus 1 end fraction plus 1 over x plus fraction numerator 1 over denominator x plus 1 end fraction less or equal than 0 end style adalah ....

  1. begin mathsize 14px style x less than negative 1 end style atau size 14px minus size 14px 1 over size 14px 3 square root of size 14px 3 size 14px less or equal than size 14px x size 14px less than size 14px 0 atau size 14px 1 over size 14px 3 square root of size 14px 3 size 14px less or equal than size 14px x size 14px less than size 14px 1 

  2. begin mathsize 14px style x less or equal than negative 1 end style atau begin mathsize 14px style negative 1 third square root of 3 less or equal than x less or equal than 0 end style atau size 14px 1 over size 14px 3 square root of size 14px 3 size 14px less or equal than size 14px x size 14px less or equal than size 14px 1

  3. begin mathsize 14px style negative 1 less than x less or equal than negative 1 third square root of 3 end style atau 0 less than x less or equal than 1 third square root of 3 atau size 14px x size 14px greater than size 14px 1

  4. begin mathsize 14px style negative 1 less or equal than x less or equal than negative 1 third square root of 3 end style atau size 14px 0 size 14px less or equal than size 14px x size 14px less or equal than size 14px 1 over size 14px 3 square root of size 14px 3 atau size 14px x size 14px greater or equal than size 14px 1

  5. begin mathsize 14px style negative 1 less or equal than x less than negative 1 third square root of 3 end style atau size 14px 0 size 14px less or equal than size 14px x size 14px less than size 14px 1 over size 14px 3 square root of size 14px 3 atau undefined

M. Haidar.

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Perhatikan perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator x minus 1 end fraction plus 1 over x plus fraction numerator 1 over denominator x plus 1 end fraction end cell less or equal than 0 row cell fraction numerator x open parentheses x plus 1 close parentheses over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction plus fraction numerator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction plus fraction numerator open parentheses x minus 1 close parentheses x over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction end cell less or equal than 0 row cell fraction numerator x squared plus x over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction plus fraction numerator x squared minus 1 over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction plus fraction numerator x squared minus x over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction end cell less or equal than 0 row cell fraction numerator open parentheses x squared plus x close parentheses plus open parentheses x squared minus 1 close parentheses plus open parentheses x squared minus x close parentheses over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction end cell less or equal than 0 row cell fraction numerator 3 x squared minus 1 over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction end cell less or equal than 0 end table end style 

Perhatikan bentuk pembilangnya, yaitu 3 x squared minus 1. Pembuat nol dari pembilangnya diperoleh dengan perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x squared minus 1 end cell equals 0 row cell 3 x squared minus 1 plus 1 end cell equals cell 0 plus 1 end cell row cell 3 x squared end cell equals 1 row cell fraction numerator 3 x squared over denominator 3 end fraction end cell equals cell 1 third end cell row cell x squared end cell equals cell 1 third end cell row x equals cell plus-or-minus square root of 1 third end root end cell row x equals cell plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction end cell row x equals cell plus-or-minus 1 third square root of 3 end cell end table

Jadi, pembuat nolnya adalah begin mathsize 14px style x equals 1 third square root of 3 end style atau x equals negative 1 third square root of 3.

Selanjutnya, pembuat nol pada penyebutnya diperoleh dengan perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x space minus space 1 right parenthesis x left parenthesis x space plus space 1 right parenthesis end cell equals 0 row cell x text = end text 1 comma space x text = end text 0 comma space atau space x end cell equals cell negative 1 end cell end table

Kemudian, perhatikan garis bilangan berikut ini!

Perhatikan bahwa ketika x equals negative 1 comma x equals 0 comma atau x equals 1 comma maka penyebutnya menjadi nol.

Oleh sebab itu, ketika x equals negative 1 comma x equals 0 comma atau x equals 1 comma maka fraction numerator 3 x squared minus 1 over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction tidak terdefinisi.

Selanjutnya, ketika begin mathsize 14px style x equals negative 1 third square root of 3 end style atau x equals 1 third square root of 3 comma maka begin mathsize 14px style fraction numerator 3 x squared minus 1 over denominator open parentheses x minus 1 close parentheses x open parentheses x plus 1 close parentheses end fraction end style  bernilai 0.

Jadi, didapat garis bilangan sebagai berikut.

Karena tanda pertidaksamaannya adalah less or equal than comma maka pilih daerah yang nilainya negatif atau nol. Oleh karena itu, didapat penyelesaian begin mathsize 14px style x less than negative 1 end style atau size 14px minus size 14px 1 over size 14px 3 square root of size 14px 3 size 14px less or equal than size 14px x size 14px less than size 14px 0 atau 1 third square root of 3 less or equal than x less than 1.

Jadi, jawaban yang tepat adalah A.

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