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Tentukan himpunan penyelesaian dari persamaan beri...

Tentukan himpunan penyelesaian dari persamaan berikut open vertical bar x minus 3 close vertical bar plus open vertical bar 2 x minus 8 close vertical bar equals 5.

Jawaban:

Persamaan nilai mutlak adalah suatu nilai mutlak dari sebuah bilangan yang dapat didefinisikan sebagai jarak bilangan tersebut terhadap titik 0 pada garis bilangan tanpa memperhatikan arahnya. Ingat definisi nilai mutlak x yaitu open vertical bar x close vertical bar equals left curly bracket subscript negative x blank jika blank straight x blank less than 0 end subscript superscript x blank jika blank straight x blank greater or equal than 0 end superscript sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 3 close vertical bar end cell rightwards arrow cell x minus 3 equals 0 end cell row x equals 3 end table

didapat open vertical bar x minus 3 close vertical bar equals left curly bracket subscript negative left parenthesis x minus 3 right parenthesis blank jika blank straight x blank less than 3 end subscript superscript x minus 3 blank jika blank straight x blank greater or equal than 3 end superscript blank dan

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar 2 x minus 8 close vertical bar end cell rightwards arrow cell 2 x minus 8 equals 0 end cell row cell 2 x end cell equals 8 row x equals cell 4 over 2 end cell row x equals 4 end table

didapat open vertical bar 2 x minus 8 close vertical bar equals left curly bracket subscript negative left parenthesis 2 x minus 8 right parenthesis blank jika blank straight x blank less than 4 end subscript superscript 2 x minus 8 blank jika blank straight x blank greater or equal than 4 end superscript.

Maka untuk straight x blank less than 3

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 3 close vertical bar plus open vertical bar 2 x minus 8 close vertical bar end cell equals 5 row cell negative open parentheses x minus 3 close parentheses minus open parentheses 2 x minus 8 close parentheses end cell equals 5 row cell negative x plus 3 minus 2 x plus 8 end cell equals 5 row cell negative 3 x plus 11 end cell equals 5 row cell negative 3 x end cell equals cell 5 minus 11 end cell row cell negative 3 x end cell equals cell negative 6 end cell row x equals cell fraction numerator negative 6 over denominator negative 3 end fraction end cell row x equals 2 end table

karena straight x less than 3, maka x equals 2 blank memenuhi.

Untuk 3 less or equal than straight x less than 4

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 3 close vertical bar plus open vertical bar 2 x minus 8 close vertical bar end cell equals 5 row cell open parentheses x minus 3 close parentheses minus open parentheses 2 x minus 8 close parentheses end cell equals 5 row cell x minus 3 minus 2 x plus 8 end cell equals 5 row cell negative x plus 5 end cell equals 5 row cell negative x end cell equals cell 5 minus 5 end cell row cell negative x end cell equals 0 row x equals 0 end table

karena 3 less or equal than straight x less than 4, maka x equals 0 blanktidak memenuhi.

Untuk straight x greater or equal than 4

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x minus 3 close vertical bar plus open vertical bar 2 x minus 8 close vertical bar end cell equals 5 row cell open parentheses x minus 3 close parentheses plus open parentheses 2 x minus 8 close parentheses end cell equals 5 row cell x minus 3 plus 2 x minus 8 end cell equals 5 row cell 3 x minus 11 end cell equals 5 row cell 3 x end cell equals cell 5 plus 11 end cell row cell 3 x end cell equals 16 row x equals cell 16 over 3 end cell end table

karena straight x greater or equal than 4, maka x equals 16 over 3 memenuhi.

Didapat nilai x yang memenuhi yaitu 2 atau 16 over 3.

Jadi, himpunan penyelesaian dari persamaan berikut open vertical bar x minus 3 close vertical bar plus open vertical bar 2 x minus 8 close vertical bar equals 5 adalah open curly brackets 2 comma 16 over 3 close curly brackets.

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