Materi Matematika Wajib : Pertidaksamaan Rasional dan Irasional

Materi Matematika Wajib : Pertidaksamaan Rasional dan Irasional

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A. Pertidaksamaan Polinomial Satu Variabel
Hasil gambar untuk math aesthetic



\begin{array}{ll}\\ \underline{\textbf{Bentuk Umum}}&:\\ &\begin{cases} a\left ( x-x_{1} \right )\left ( x-x_{2} \right )\left ( x-x_{3} \right )...< 0 \\ a\left ( x-x_{1} \right )\left ( x-x_{2} \right )\left ( x-x_{3} \right )... \leq 0 \\ a\left ( x-x_{1} \right )\left ( x-x_{2} \right )\left ( x-x_{3} \right )... > 0 \\ a\left ( x-x_{1} \right )\left ( x-x_{2} \right )\left ( x-x_{3} \right )... \geq 0 \end{cases}\\ & \end{array}.
A. 1 Bentuk Linear
Sudah dipelajari di tingkat sebelumnya
A. 2 Bentuk Kuadrat
\begin{array}{ll}\\ \underline{\textbf{Bentuk Umum}}&:\\ &\begin{cases} ax^{2}+bx+c< 0 \\ ax^{2}+bx+c\leq 0 \\ ax^{2}+bx+c > 0 \\ ax^{2}+bx+c \geq 0 \end{cases}\\ & \end{array}.
A. 2. 1 Penyelesaian Bentuk Kuadrat
\begin{array}{|l|c|}\hline \multicolumn{2}{|c|}{\begin{aligned}&\\ ax^{2}+bx+c\: \: ...\: \: 0&\quad \: \textrm{diubah menjadi}\quad \: ax^{2}+bx+c=0\\ &\Leftrightarrow a\left ( x-x_{1} \right )\left ( x-x_{2} \right )=0\\ &\Leftrightarrow x=x_{1}\quad \textrm{atau}\quad x=x_{2}\\ &\end{aligned}}\\\hline \textrm{Pertidaksamaan}&\textrm{Himpunan Penyelesaian dengan}\: \:x_{1}<x_{2} \\\hline ax^{2}+bx+c< 0&\left \{ x|x_{1}<x<x_{2},\: x\in \mathbb{R} \right \} \\\hline ax^{2}+bx+c\leq 0&\left \{ x|x_{1}\leq x\leq x_{2},\: x\in \mathbb{R} \right \} \\\hline ax^{2}+bx+c> 0&\left \{ x|x<x_{1}\: \: \textrm{atau}\: \: x>x_{2},\: x\in \mathbb{R} \right \} \\\hline ax^{2}+bx+c\geq 0&\left \{ x|x\leq x_{1}\: \: \textrm{atau}\: \: x\geq x_{2},\: x\in \mathbb{R} \right \} \\\hline \end{array}.
\LARGE\fbox{\LARGE\fbox{CONTOH SOAL}}.
\begin{array}{ll}\\ \fbox{1}.&\textrm{Tentukan himpunan penyelesaian \textbf{PtKSV}(Pertidaksamaan Linear Satu Variabel)}\\ &\textrm{a}.\quad x^{2}-6x+8< 0\\ &\textrm{b}.\quad x^{2}-6x+8\leq 0\\ &\textrm{c}.\quad x^{2}-6x+8> 0\\ &\textrm{d}.\quad x^{2}-6x+8\geq 0\\\\ &\textrm{Jawab}:\\ \end{array}.
\begin{array}{ll}\\ . \quad .&\begin{array}{|l|c|c|}\hline \multicolumn{3}{|c|}{\begin{aligned}&\\ x^{2}-6x+8\: \: ...\: \: 0&\quad \: \textrm{diubah menjadi}\quad \: x^{2}-6x+8=0\\ &\Leftrightarrow 1\left ( x-2 \right )\left ( x-4 \right )=0\\ &\Leftrightarrow x=2\quad \textrm{atau}\quad x=4\\ &\end{aligned}}\\\hline \textrm{Pertidaksamaan}&\textrm{HP dengan}\: \:\left (2<4 \right )&\textrm{Selang/Interval}\\\hline x^{2}-6x+8< 0&\left \{ x|2<x<4,\: x\in \mathbb{R} \right \}&\begin{array}{ll|llll|llll}\\ &\multicolumn{2}{c}{.}&&&\multicolumn{2}{c}{.}&\\ &\multicolumn{2}{r}{.}&&&\multicolumn{2}{l}{.}&&\\\cline{3-6} &+&&-&-&&+&+&\\\hline &\multicolumn{2}{r}{2}&&&\multicolumn{2}{l}{4}&& \end{array} \\\hline x^{2}-6x+8\leq 0&\left \{ x|2\leq x\leq 4,\: x\in \mathbb{R} \right \}&\begin{array}{ll|llll|llll}\\ &\multicolumn{2}{c}{.}&&&\multicolumn{2}{c}{.}&\\ &\multicolumn{2}{r}{.}&&&\multicolumn{2}{l}{.}&&\\\cline{3-6} &+&&-&-&&+&+&\\\hline &\multicolumn{2}{r}{\textcircled{2}}&&&\multicolumn{2}{l}{\textcircled{4}}&& \end{array} \\\hline x^{2}-6x+8> 0&\left \{ x|x<2\: \: \textrm{atau}\: \: x>4,\: x\in \mathbb{R} \right \}&\begin{array}{ll|llll|llll}\\ &\multicolumn{2}{c}{.}&&&\multicolumn{2}{c}{.}&\\ &\multicolumn{2}{r}{.}&&&\multicolumn{2}{l}{.}&&\\\cline{1-2}\cline{7-9} &+&&-&-&&+&+&\\\hline &\multicolumn{2}{r}{2}&&&\multicolumn{2}{l}{4}&& \end{array} \\\hline x^{2}-6x+8\geq 0&\left \{ x|x\leq 2\: \: \textrm{atau}\: \: x\geq 4,\: x\in \mathbb{R} \right \}&\begin{array}{ll|llll|llll}\\ &\multicolumn{2}{c}{.}&&&\multicolumn{2}{c}{.}&\\ &\multicolumn{2}{r}{.}&&&\multicolumn{2}{l}{.}&&\\\cline{1-2}\cline{7-9} &+&&-&-&&+&+&\\\hline &\multicolumn{2}{r}{\textcircled{2}}&&&\multicolumn{2}{l}{\textcircled{4}}&& \end{array} \\\hline \multicolumn{3}{|c|}{\begin{aligned}\textbf{Catatan: }&\textrm{Angka yang dilingkari termasuk}\\ &\textrm{himpunan penyelesaian} \end{aligned}}\\\hline \end{array} \end{array}.

Sumber Referensi
  1. Kanginan, M. 2016. Matematika untuk SMA-MA/SMK-SAK Kelas X (Edisi Revisi 2016). Bandung: SEWU

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