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Tentukan interval nilai x yang memenuhi pertidaksamaan berikut: 9. ∣∣​x2−x−10∣∣​≤10

Pertanyaan

Tentukan interval nilai x yang memenuhi pertidaksamaan berikut:

9. open vertical bar x squared minus x minus 10 close vertical bar less or equal than 10

R. Bella

Master Teacher

Mahasiswa/Alumni UIN Syarif Hidayatullah Jakarta

Jawaban terverifikasi

Jawaban

 interval yang memenuhi adalah negative 4 less or equal than x less or equal than 0 space logical or space 1 less or equal than x less or equal than 5

Pembahasan

Ingat, sifat pertidaksamaan nilai mutlak:

open vertical bar f left parenthesis x right parenthesis close vertical bar less or equal than c left right arrow negative c less or equal than f left parenthesis x right parenthesis less or equal than c

Berdasarkan sifat tersebut diperoleh:

open vertical bar x squared minus x minus 10 close vertical bar less or equal than 10 left right arrow negative 10 less or equal than x squared minus x minus 10 less or equal than 10

Sehingga diperoleh perhitungan untuk ruas kiri:

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus x minus 10 end cell less or equal than cell negative 10 end cell row cell x squared minus x end cell less or equal than 0 row cell x left parenthesis x minus 1 right parenthesis end cell less or equal than 0 row x equals cell 0 comma space x equals 1 end cell end table

Sehingga diperoleh perhitungan untuk ruas kanan:

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus x minus 10 end cell greater or equal than 10 row cell x squared minus x minus 20 end cell greater or equal than 0 row cell left parenthesis x minus 5 right parenthesis left parenthesis x plus 4 right parenthesis end cell greater or equal than 0 row x equals cell 5 comma space x equals negative 4 end cell end table

Diperoleh lima daerah penyelesaian:

table attributes columnalign right center left columnspacing 0px end attributes row x less or equal than cell negative 4 end cell row cell negative 4 end cell less or equal than cell x less or equal than 0 end cell row 0 less or equal than cell x less or equal than 1 end cell row 1 less or equal than cell x less or equal than 5 end cell row x greater or equal than 5 end table

Dengan menggunakan uji titik diperoleh interval yang memenuhi adalah negative 4 less or equal than x less or equal than 0 space logical or space 1 less or equal than x less or equal than 5.

Jadi, interval yang memenuhi adalah negative 4 less or equal than x less or equal than 0 space logical or space 1 less or equal than x less or equal than 5

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