Contoh Soal 3 Pertidaksamaan Rasional dan Irasional Satu Variabel (Kelas X Matematika Wajib)

$\begin{array}{ll}\\ 11.&\textrm{Penyelesaian pertidaksamaan}\\ &\displaystyle \sqrt{6x-5}\leq x \: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \textrm{a}.&1<x<0\\\\ \textrm{b}.&x<1\: \: \textrm{atau}\: \: x\geq 5\\\\ \color{red}\textrm{c}.&\displaystyle \frac{5}{6}\leq x\leq 1\: \: \textrm{atau}\: \: x\geq 5\\\\ \textrm{d}.&\displaystyle \frac{5}{6}\leq x< 1\: \: \textrm{atau}\: \: 5<x<6\\\\ \textrm{e}.&x\geq 6 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\sqrt{6x-5}&\leq x\\ 1.\quad\textrm{Kuadrat}&\textrm{kan}\\ 6x-5&\leq x^{2}\\ -x^{2}+6x-5&\leq 0\\ x^{2}-6x+5&\geq 0\\ (x-1)(x-5)&\geq 0\\ x\leq 1\: \: \textrm{atau}&\: \: x\geq 5\\ 2.\quad \textrm{Di bawa}&\textrm{h tanda akar}\: \: \geq 0\\ 6x-5&\geq 0\\ 6x&\geq 5\\ x&\geq \displaystyle \frac{5}{6} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 12.&\textrm{Penyelesaian pertidaksamaan}\\ &\displaystyle \sqrt{6x+6}>6 \: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&x>7\\ \textrm{b}.&x\geq 7\\ \textrm{c}.&x<7\\ \textrm{d}.&x>1\\ \textrm{e}.&x\geq 1 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\sqrt{6x+6}&>6\\ 1.\quad\textrm{Kuadrat}&\textrm{kan}\\ 6x+6&>36\\ x+1&>6\\ x&>7\\ 2.\quad \textrm{Di bawa}&\textrm{h tanda akar}\: \: \geq 0\\ 6x+6&\geq 0\\ 6x&\geq 6\\ x&\geq \displaystyle \frac{6}{6}\\ x&\geq 1 \end{aligned} \end{array}$

$\begin{array}{l}\\ 13.&\textrm{Penyelesaian pertidaksamaan}\\ &x+2>\displaystyle \sqrt{10-x^{2}} \: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \textrm{a}.&2\leq x\leq \sqrt{10}\\ \color{red}\textrm{b}.&1<x\leq \sqrt{10}\\ \textrm{c}.&-3<x\leq \sqrt{10}\\ \textrm{d}.&-\sqrt{10}\leq x\leq \sqrt{10}\\ \textrm{e}.&x< -3\: \: \textrm{atau}\: \: x>1 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}x+2&>\displaystyle \sqrt{10-x^{2}} \\ 1.\quad\textrm{Kuadrat}&\textrm{kan}\\ x^{2}+4x+4&>10-x^{2}\\ 2x^{2}+4x&+4-10>0\\ 2x^{2}+4x-&6>0\\ x^{2}+2x-&3>0\\ (x+3)&(x-1)>0\\ x<-3&\: \: \textrm{atau}\: \: x>1\\ 2.\quad \textrm{Di bawa}&\textrm{h tanda akar}\: \: \geq 0\\ 10-x^{2}&\geq 0\\ x^{2}-10&\leq 0\\ (x-\sqrt{10})&(x+\sqrt{10})\leq 0\\ -\sqrt{10}\leq x&\leq \sqrt{10} \end{aligned} \end{array}$

$\begin{array}{l}\\ 14.&\textrm{Penyelesaian pertidaksamaan}\\ &\displaystyle \sqrt{3x+7}\geq x-1 \: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \textrm{a}.&-1<x<6\\ \textrm{b}.&-1\leq x<6\\ \textrm{c}.&x\geq -\displaystyle \frac{7}{3}\\ \color{red}\textrm{d}.&-\displaystyle \frac{7}{3}\leq x\leq 6\\ \textrm{e}.&-\displaystyle \frac{7}{3}\leq x\leq 1 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\sqrt{3x+7}&\geq x-1 \\ 1.\quad\textrm{Kuadrat}&\textrm{kan}\\ 3x+7&\geq x^{2}-2x+1\\ -x^{2}+3x&+2x+7-1\geq 0\\ -x^{2}+5x+&6\geq 0\\ x^{2}-5x-&6\leq 0\\ (x+1)&(x-6)\leq 0\\ -1\leq x&\leq 6\\ 2.\quad \textrm{Di bawa}&\textrm{h tanda akar}\: \: \geq 0\\ 3x+7&\geq 0\\ 3x&\geq -7\\ x&\geq -\displaystyle \frac{7}{3} \end{aligned} \end{array}$

 $\begin{array}{ll}\\ 15.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ & \sqrt{2x-8}<\sqrt{x+5}\: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \textrm{a}.&x\geq -5\\ \textrm{b}.&x<-13\: \: \textrm{atau}\: \: x\geq -4\\ \textrm{c}.&x<13\\ \color{red}\textrm{d}.&4\leq x< 13\\ \textrm{e}.&-5\leq x\leq 4 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\sqrt{2x-8}&<\sqrt{x+5}\\ (1)\: \: \, \textrm{kuadratkan}&\\ 2x-8&<x+5\\ x&<13\\ (2)\quad 2x-8\geq 0&\\ x&\geq 4\\ (3)\: \: \quad x+5\geq 0&\\ x&\geq 5\\ \textrm{perhatikan}&\textrm{lah garis bilangannya berikut}\\ 1 \qquad&\color{black}\begin{array}{ccccccc|ccccc}\\ &&&&&&&&&&\\\cline{1-7} &&&&&&&&&&\textrm{X}\\\hline &&&&&&13&&&& \end{array}\\\\ 2\qquad&\color{black}\begin{array}{ccc|ccccccccc}\\ &&&&&&&&&&\\\cline{4-11} &&&&&&&&&&\textrm{X}\\\hline &&&4&&&&&&& \end{array}\\\\ 3\qquad&\color{black}\begin{array}{ccccc|ccccccc}\\ &&&&&&&&&&\\\cline{6-11} &&&&&&&&&&\textrm{X}\\\hline &&&&&5&&&&& \end{array} \\ \end{aligned} \end{array}$