Pertanyaan

Himpunan penyelesaian pertidaksamaan x 1 > x + 2 x adalah ....

Himpunan penyelesaian pertidaksamaan $\frac{1}{x} > \frac{x}{x + 2}$ adalah ....

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Jawaban terverifikasi

Pembahasan

Nilai x yang memenuhi adalah Buat garis bilangan untuk nilai x yang memenuhi, sehingga : Uji masing-masing daerah menjadi : Maka garis bilangan menjadi : Karena pertidaksamaan berbentuk “kurang dari” maka daerah yang dipilih adalah daerah negatif sehingga himpunan penyelesaiannya adalah

Nilai x yang memenuhi adalah

$1 over straight x greater than fraction numerator straight x over denominator straight x plus 2 end fraction 1 over straight x minus fraction numerator straight x over denominator straight x plus 2 end fraction greater than 0 fraction numerator left parenthesis straight x plus 2 right parenthesis minus straight x squared over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction greater than 0 space space space space space left parenthesis kedua space ruas space dikali space left parenthesis negative 1 right parenthesis right parenthesis fraction numerator straight x squared minus straight x minus 2 over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction less than 0 fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction less than 0 straight x equals negative 2 semicolon space straight x equals negative 1 semicolon space straight x equals 0 space atau space straight x equals 2 Syarat colon space straight x not equal to negative 2 space dan space straight x not equal to 0$

Buat garis bilangan untuk nilai x yang memenuhi, sehingga :

Uji masing-masing daerah menjadi :

$Untuk space straight x less than negative 2 comma space pilih space straight x equals negative 3 space menjadi fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis left parenthesis negative 3 right parenthesis plus 1 right parenthesis left parenthesis left parenthesis negative 3 right parenthesis minus 2 right parenthesis over denominator left parenthesis negative 3 right parenthesis left parenthesis left parenthesis negative 3 right parenthesis plus 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis negative 2 right parenthesis left parenthesis negative 5 right parenthesis over denominator left parenthesis negative 3 right parenthesis left parenthesis negative 1 right parenthesis end fraction equals 10 over 3 space left parenthesis positif right parenthesis Untuk space minus 2 less than straight x less than negative 1 comma space pilih space straight x equals negative 3 over 2 space menjadi fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis left parenthesis negative begin display style 3 over 2 end style right parenthesis plus 1 right parenthesis left parenthesis left parenthesis negative begin display style 3 over 2 end style right parenthesis minus 2 right parenthesis over denominator left parenthesis negative begin display style 3 over 2 end style right parenthesis left parenthesis left parenthesis negative begin display style 3 over 2 end style right parenthesis plus 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis negative begin display style 1 half end style right parenthesis left parenthesis negative begin display style 7 over 2 end style right parenthesis over denominator left parenthesis negative begin display style 3 over 2 end style right parenthesis left parenthesis begin display style 1 half end style right parenthesis end fraction equals negative 7 over 3 space left parenthesis negatif right parenthesis Untuk space minus 1 less than straight x less than 0 comma space pilih space straight x equals negative 1 half space menjadi fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis left parenthesis negative begin display style 1 half end style right parenthesis plus 1 right parenthesis left parenthesis left parenthesis negative begin display style 1 half end style right parenthesis minus 2 right parenthesis over denominator left parenthesis negative begin display style 1 half end style right parenthesis left parenthesis left parenthesis negative begin display style 1 half end style right parenthesis plus 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis begin display style 1 half end style right parenthesis left parenthesis negative begin display style 5 over 2 end style right parenthesis over denominator left parenthesis negative begin display style 1 half end style right parenthesis left parenthesis begin display style 3 over 2 end style right parenthesis end fraction equals 5 over 3 space left parenthesis positif right parenthesis Untuk space 0 less than straight x less than 2 comma space pilih space straight x equals 1 space menjadi fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis left parenthesis 1 right parenthesis plus 1 right parenthesis left parenthesis left parenthesis 1 right parenthesis minus 2 right parenthesis over denominator left parenthesis 1 right parenthesis left parenthesis left parenthesis 1 right parenthesis plus 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 2 right parenthesis left parenthesis negative 1 right parenthesis over denominator left parenthesis 1 right parenthesis left parenthesis 3 right parenthesis end fraction equals negative 2 over 3 space left parenthesis negatif right parenthesis Untuk space straight x greater than 2 comma space pilih space straight x equals 3 space menjadi fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator straight x left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis left parenthesis 3 right parenthesis plus 1 right parenthesis left parenthesis left parenthesis 3 right parenthesis minus 2 right parenthesis over denominator left parenthesis negative 3 right parenthesis left parenthesis left parenthesis negative 3 right parenthesis plus 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 4 right parenthesis left parenthesis 1 right parenthesis over denominator left parenthesis 3 right parenthesis left parenthesis 5 right parenthesis end fraction equals 4 over 15 space left parenthesis positif right parenthesis$

Maka garis bilangan menjadi :