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Pertanyaan

Jika x bilangan bulat, tentukan himpunan penyelesaian dari pertidaksamaan berikut a. 2 ( 2 x − 1 ) < 3 ( 2 x + 2 )

Jika  bilangan bulat, tentukan himpunan penyelesaian dari pertidaksamaan berikut 

a.  

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D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

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himpunan penyelesaian dari pertidaksamaan tersebut adalah HP = { x ∣ x > − 4 , x ∈ bilangan bulat } .

himpunan penyelesaian dari pertidaksamaan tersebut adalah 

Pembahasan

Kita dapat menyelesaikan pertidaksamaan linear satu variabel dengan cara menambah, mengurang, mengali, atau membagi kedua ruas dengan bilangan yang sama. Khusus untuk perkalian dan pembagian kedua ruas dengan bilangan negatif, tanda pertidaksamaannya berubah/dibalik. Ingat kembali sifat distributif a ( b + c ) = ab + a c Perhatikan perhitungan berikut! &\frac8{{\color[rgb]{0.0, 0.0, 1.0}-}{\color[rgb]{0.0, 0.0, 1.0}2}}\\x&>&-4\end{array}" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mn»2«/mn»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»3«/mn»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«mo»+«/mo»«mn mathcolor=¨#0000FF¨»2«/mn»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«mo»+«/mo»«mn mathcolor=¨#0000FF¨»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn mathcolor=¨#0000FF¨»6«/mn»«mi mathcolor=¨#0000FF¨»x«/mi»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«mo»-«/mo»«mn mathcolor=¨#0000FF¨»6«/mn»«mi mathcolor=¨#0000FF¨»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»§#60;«/mo»«/mtd»«mtd»«mn»8«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«mrow»«mo mathcolor=¨#0000FF¨»-«/mo»«mn mathcolor=¨#0000FF¨»2«/mn»«/mrow»«/mfrac»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mfrac»«mn»8«/mn»«mrow»«mo mathcolor=¨#0000FF¨»-«/mo»«mn mathcolor=¨#0000FF¨»2«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»§#62;«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/math»" role="math" src="data:image/png;base64,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" style="max-width: none;"> Dengan demikian, himpunan penyelesaian dari pertidaksamaan tersebut adalah HP = { x ∣ x > − 4 , x ∈ bilangan bulat } .

Kita dapat menyelesaikan pertidaksamaan linear satu variabel dengan cara menambah, mengurang, mengali, atau membagi kedua ruas dengan bilangan yang sama. Khusus untuk perkalian dan pembagian kedua ruas dengan bilangan negatif, tanda pertidaksamaannya berubah/dibalik.

Ingat kembali sifat distributif

 

Perhatikan perhitungan berikut! 

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses 2 x minus 1 close parentheses end cell less than cell 3 open parentheses 2 x plus 2 close parentheses end cell row cell 4 x minus 2 end cell less than cell 6 x plus 6 end cell row cell 4 x minus 2 plus 2 end cell less than cell 6 x plus 6 plus 2 end cell row cell 4 x end cell less than cell 6 x plus 8 end cell row cell 4 x minus 6 x end cell less than cell 6 x plus 8 minus 6 x end cell row cell negative 2 x end cell less than 8 row cell fraction numerator negative 2 x over denominator negative 2 end fraction end cell greater than cell fraction numerator 8 over denominator negative 2 end fraction end cell row x greater than cell negative 4 end cell end table  

Dengan demikian, himpunan penyelesaian dari pertidaksamaan tersebut adalah 

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