Question

সমাধান সেট নির্ণয় কর: $\frac{1}{x+2}+\frac{1}{x+5}=\frac{1}{x+3}+\frac{1}{x+4}$

Answer 1

$\frac{1}{x+2}+\frac{1}{x+5}=\frac{1}{x+3}+\frac{1}{x+4}$

বা, $\frac{1}{x+2}-\frac{1}{x+3}=\frac{1}{x+4}-\frac{1}{x+5}$

বা, $\frac{\left(x+3\right)-\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}=\frac{\left(x+5\right)-\left(x+4\right)}{\left(x+4\right)\left(x+5\right)}$

বা, $\frac{x+3-x+2}{\left(x+2\right)\left(x+3\right)}=\frac{x+5-x+4}{\left(x+4\right)\left(x+5\right)}$

বা, $\frac{1}{\left(x+2\right)\left(x+3\right)}=\frac{1}{\left(x+4\right)\left(x+5\right)}$

বা, $\left(x+4\right)\left(x+5\right)=\left(x+2\right)\left(x+3\right)$

বা, $x\left(x+5\right)+4\left(x+5\right)=x\left(x+3\right)+2\left(x+3\right)$

বা, $x^2+5x+4x+20=x^2+3x+2x+6$

বা, $x^2+9x+20=x^2+5x+6$

বা, $x^2+9x+20-x^2-5x-6=0$

বা, $4x+14=0$

বা, $4x=-14$

বা, $x=\frac{-14}{4}$

$\therefore x=-\frac72$

সুতরাং নির্ণেয় সমাধান সেট $S=\left\{-\frac72\right\}$ [Answer]