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Himpunan penyelesaian dari ∣x−1∣+∣2x∣+∣3x+1∣=6 adalah ...

Pertanyaan

Himpunan penyelesaian dari open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar equals 6 adalah ...

  1. open curly brackets negative 4 comma 4 close curly brackets

  2. open curly brackets negative 1 comma 3 close curly brackets

  3. open curly brackets negative 4 comma negative 1 close curly brackets

  4. open curly brackets 1 comma 2 close curly brackets

  5. open curly brackets negative 1 comma 1 close curly brackets

M. Mariyam

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Jawaban

pilihan jawaban yang tepat adalah E.

Pembahasan

Ingat kembali konsep definisi nilai mutlak secara matematis sebagai berikut:

open vertical bar f open parentheses x close parentheses close vertical bar equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell f open parentheses x close parentheses end cell cell semicolon f open parentheses x close parentheses greater or equal than 0 end cell row cell negative f open parentheses x close parentheses end cell cell semicolon f open parentheses x close parentheses less than 0 end cell end table close

Oleh karena itu,  open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar equals 6 dapat diselesaikan dengan beberapa langkah:

Berdasarkan definisi nilai mutlak, maka

open vertical bar x minus 1 close vertical bar equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell x minus 1 end cell cell comma space x greater or equal than 1 end cell row cell 1 minus x end cell cell comma space x less than 1 end cell end table close

open vertical bar 2 x close vertical bar equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell 2 x end cell cell comma space x greater or equal than 0 end cell row cell negative 2 x end cell cell comma space x less than 0 end cell end table close

open vertical bar 3 x plus 1 close vertical bar equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell 3 x plus 1 end cell cell comma space x greater or equal than negative 1 third end cell row cell negative 3 x minus 1 end cell cell comma space x less than negative 1 third end cell end table close

Irisan dari semua interval terbagi menjadi 4 bagian, yaitu

x less than negative 1 thirdnegative 1 third less or equal than x less than 00 less or equal than x less than 1, dan x greater or equal than 1

Untuk x less than negative 1 third,

 table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar end cell equals 6 row cell negative open parentheses x minus 1 close parentheses plus open parentheses negative 2 x close parentheses plus open parentheses negative 3 x minus 1 close parentheses end cell equals 6 row cell negative x plus 1 minus 2 x minus 3 x minus 1 end cell equals 6 row cell negative 6 x end cell equals 6 row x equals cell negative 1 end cell end table

Untuk negative 1 third less or equal than x less than 0

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar end cell equals 6 row cell negative open parentheses x minus 1 close parentheses plus open parentheses negative 2 x close parentheses plus 3 x plus 1 end cell equals 6 row cell negative x plus 1 minus 2 x plus 3 x plus 1 end cell equals 6 row 2 equals 6 end table

tidak ada penyelesaian di interval negative 1 third less or equal than x less than 0.

Untuk 0 less or equal than x less than 1,

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar end cell equals 6 row cell negative open parentheses x minus 1 close parentheses plus 2 x plus 3 x plus 1 end cell equals 6 row cell negative x plus 1 plus 2 x plus 3 x plus 1 end cell equals 6 row cell 4 x plus 2 end cell equals 6 row cell 4 x end cell equals 4 row x equals 1 end table

Untuk  x greater or equal than 1,

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar end cell equals 6 row cell x minus 1 plus 2 x plus 3 x plus 1 end cell equals 6 row cell x minus 1 plus 2 x plus 3 x plus 1 end cell equals 6 row cell 6 x end cell equals 6 row x equals 1 end table

Dengan demikian, diperoleh himpunan penyelesaian dari open vertical bar x minus 1 close vertical bar plus open vertical bar 2 x close vertical bar plus open vertical bar 3 x plus 1 close vertical bar equals 6 adalah open curly brackets negative 1 comma 1 close curly brackets.

Jadi, pilihan jawaban yang tepat adalah E.

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