Question

SOLVE 1 divided by (x-1)(x-2)+1 divided by (x-2)(x-3)=2 divided by 3

Answer 1

$$\begin{gathered} \frac{1}{\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)}+\frac{1}{\left(\mathrm{x}-2\right)\left(\mathrm{x}-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{\left(\mathrm{x}-3\right)+\left(\mathrm{x}-1\right)}{\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\left(\mathrm{x}-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{2\mathrm{x}-4}{\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\left(\mathrm{x}-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{2\left(\mathrm{x}-2\right)}{\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\left(\mathrm{x}-3\right)}=\frac{2}{3} \\ \Rightarrow \left(\mathrm{x}-1\right)\left(\mathrm{x}-3\right)=3 \\ \Rightarrow {\mathrm{x}}^2-4\mathrm{x}+3=3 \\ \Rightarrow {\mathrm{x}}^2-4\mathrm{x}=0 \\ \Rightarrow \mathrm{x}\left(\mathrm{x}-4\right)=0 \\ \Rightarrow \mathrm{x}=0\mathrm{or}\mathrm{x}=4 \end{gathered}$$