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استعمل الجمع والطرح لحل كل من المتباينات التالية في Z:

$\begin{array}{rl}& x+1<8\end{array}$

$x+1-1<8-1\Rightarrow x<7$

$\begin{array}{rl}& x-12\ge 24\end{array}$

$x-12+12\ge 24+12\Rightarrow X\ge 36$

$\begin{array}{rl}& 6+z\le 34\end{array}$

$6-6+z\le 34-6\Rightarrow z\le 28$

$\begin{array}{rl}& y+42>0\end{array}$

$\begin{array}{rl}& y+42-42>0-42\\ & y>-42\end{array}$

استعمل الضرب والقسمة لحل كل من المتباينات الآتية في Q:

$\begin{array}{rl}& \frac{x}{15}>\frac{1}{7}\end{array}$

$\begin{array}{rl}& \frac{x}{15}\times \left(15\right)>\frac{1}{7}\times \left(15\right)\\ & x>\frac{15}{7}\end{array}$

$\begin{array}{rl}& -9z\le 63\end{array}$

$\frac{9z}{-9}\ge \frac{63}{-9}\Rightarrow z\ge -7$

$\begin{array}{rl}& 2x<\frac{1}{5}\end{array}$

$\begin{array}{rl}& \frac{2x}{2}<\frac{1}{5}÷2\Rightarrow x<\frac{1}{5}\times \frac{1}{2}\\ & x<\frac{1}{10}\end{array}$

$\begin{array}{rl}& 3y\ge -31\end{array}$

$\frac{3y}{3}\ge \frac{-31}{3}\Rightarrow y\ge \frac{-31}{3}$

استعمل خصائص المتباينات لحل كل من المتباينات الآتية:

$\begin{array}{rl}& 6\mathrm{x}+14<50,\mathrm{x}\in \mathrm{Z}\end{array}$

$$\begin{gathered} 6x+14-14<50-14 \\ \begin{array}{rl}& 6x<36\\ & \frac{6x}{6}<\frac{36}{6}\Rightarrow x<6\end{array} \end{gathered}$$

$\begin{array}{rl}& -3(\mathrm{y}-8)\ge 39,\mathrm{y}\in \mathrm{Q}\end{array}$

$$\begin{gathered} -3y+24\ge 39 \\ -3y+24-24\ge 39-24 \\ -3y\ge 15 \\ \frac{-3y}{-3}\le \frac{15}{-3} \\ y\le -5 \end{gathered}$$

$\begin{array}{rl}& 6(2\mathrm{z}+4)>2\mathrm{z},\mathrm{z}\in \mathrm{Q}\end{array}$

$$\begin{gathered} \begin{array}{rl}& 12z+24>2z\\ & 12z-2z+24>2z-2z\\ & 10z+24>0\end{array} \\ \begin{array}{ll}10z+24-24>0-24& \\ 10z>-24& \end{array} \\ \begin{array}{r}\frac{10z}{10}>\frac{-24}{10}\\ z>-2.4\end{array} \end{gathered}$$

$\begin{array}{rl}& 2(\mathrm{x}-9)\le \sqrt{25}-3\mathrm{x},\mathrm{x}\in \mathrm{Q}\end{array}$

$$\begin{gathered} 2x-18\le 5-3x \\ \begin{array}{rl}& 2x+3x-18<5-3x+3x\\ & 5x-18<5\end{array} \\ \begin{array}{rl}& 5x-18+18<5+18\\ & 5x<23\end{array} \\ \frac{5x}{5}<\frac{23}{5}\Rightarrow x<\frac{23}{5} \end{gathered}$$