حل المتباينات التالية في R باستعمال خواص المتباينات على الأعداد الحقيقية:

$\frac{1}{9}(x-\sqrt{2})<\frac{1}{3}(\sqrt{2}-x)$

$$\begin{gathered} \frac{1}{9}(x-\sqrt{2})<\frac{1}{3}(\sqrt{2}-x)\Rightarrow \frac{1}{9}x-\frac{\sqrt{2}}{9}<\frac{\sqrt{2}}{3}-\frac{1}{3}x \\ \begin{array}{rl}& \Rightarrow \frac{1}{9}\times +\frac{1}{3}\times <\frac{\sqrt{2}}{3}+\frac{\sqrt{2}}{9}\Rightarrow \frac{1+3}{9}\times <\frac{4\sqrt{2}}{9}\Rightarrow 4x<4\sqrt{2}\\ & x<\frac{4\sqrt{2}}{4}\Rightarrow x<\sqrt{2}\end{array} \end{gathered}$$

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حل المتباينات التالية في R باستعمال الخواص ومثله على مستقيم الأعداد:

$5(x-1)\le 8-\sqrt[3]{-8}$

$5x\le 15\Rightarrow x\le 3$

$\frac{1}{3}(\mathrm{z}-\frac{7}{2})+\frac{1}{3}\mathrm{z}\ge -\frac{23}{6}$

$4z\ge -16\Rightarrow z\ge -4$

$\frac{t}{2}>2(\frac{1}{\sqrt{16}}-t)$

$10t>2\Rightarrow t>\frac{1}{5}$

$7(\mathrm{n}-1)\le 9(1-\mathrm{n})$

$16\mathrm{n}\le 16\Rightarrow \mathrm{n}\le 1$

حل المتباينات التالية في R باستعمال خواص المتباينات على الأعداد الحقيقية:

$2(x-\sqrt{5})\le 11(1-\sqrt{5})$

$2x\le 11-9\sqrt{5}\Rightarrow x\le \frac{11-9\sqrt{5}}{2}$

$\frac{1}{2}\mathrm{y}-\sqrt[3]{-8}-\frac{3}{2}\mathrm{y}<|-7|$

$2y<5\Rightarrow y<\frac{5}{2}$

$12-\sqrt[3]{-125}\ge 6(z-1)$

$13\ge 6z\Rightarrow z\le \frac{13}{6}$

$\frac{-5}{7}(\frac{7}{3}m+\frac{14}{5})>0$

$-5m-6>0\Rightarrow m<\frac{-6}{5}$

$6(4-h)\le 7(h-5)$

$59\le 13\mathrm{h}\Rightarrow \mathrm{h}\ge \frac{59}{13}$

$\frac{-3z}{7}\ge \frac{1}{7}+z$

$-1\ge 10z\Rightarrow z\le -\frac{1}{10}$

$\frac{1}{9}(x-\sqrt{2})<\frac{1}{3}(\sqrt{2}-x)$

$$\begin{gathered} \frac{1}{9}(x-\sqrt{2})<\frac{1}{3}(\sqrt{2}-x)\Rightarrow \frac{1}{9}x-\frac{\sqrt{2}}{9}<\frac{\sqrt{2}}{3}-\frac{1}{3}x \\ \begin{array}{rl}& \Rightarrow \frac{1}{9}\times +\frac{1}{3}\times <\frac{\sqrt{2}}{3}+\frac{\sqrt{2}}{9}\Rightarrow \frac{1+3}{9}\times <\frac{4\sqrt{2}}{9}\Rightarrow 4x<4\sqrt{2}\\ & x<\frac{4\sqrt{2}}{4}\Rightarrow x<\sqrt{2}\end{array} \end{gathered}$$

$\frac{n}{6}+\frac{3}{2}>\frac{n}{3}-\frac{1}{3}$

$2+9>2\mathrm{n}-\mathrm{n}\Rightarrow \mathrm{n}<11$

$\frac{5}{\sqrt[3]{-8}}-4k\le \frac{5}{2}-k$

$-10\le 6\mathrm{k}\Rightarrow \mathrm{k}\ge \frac{-10}{6}$

$\frac{1}{2}(y-3)\ge \frac{1}{4}(\sqrt{2}+y)$

$2y-y\ge \sqrt{5}+6\Rightarrow y\ge \sqrt{5}+6$