a) \(\displaystyle{1 \over {3x - 2}} - {1 \over {3x + 2}} - {{3x - 6} \over {4 - 9{x^2}}}\)
\(\displaystyle = {1 \over {3x - 2}} - {1 \over {3x + 2}} + {{3x - 6} \over {9{x^2}-4}}\)
\(\displaystyle = {1 \over {3x - 2}} - {1 \over {3x + 2}} \) \(\displaystyle+ {{3x - 6} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}}\)
\(\displaystyle = {{3x + 2} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}}\)\(\displaystyle + {{ - \left( {3x - 2} \right)} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}} \) \(\displaystyle+ {{3x - 6} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}} \)
\(\displaystyle = {{3x + 2 - 3x + 2 + 3x - 6} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}} \)
\(\displaystyle= {{3x - 2} \over {\left( {3x + 2} \right)\left( {3x - 2} \right)}} = {1 \over {3x + 2}}\)
b) \(\displaystyle{{18} \over {\left( {x - 3} \right)\left( {{x^2} - 9} \right)}} - {3 \over {{x^2} - 6x + 9}} \)\(\displaystyle- {x \over {{x^2} - 9}}\)
\(\displaystyle = {{18} \over {{{\left( {x - 3} \right)}^2}\left( {x + 3} \right)}} + {{ - 3} \over {{{\left( {x - 3} \right)}^2}}}\) \(\displaystyle + {{ - x} \over {\left( {x + 3} \right)\left( {x - 3} \right)}}\)
\(\displaystyle = {{18} \over {{{\left( {x - 3} \right)}^2}\left( {x + 3} \right)}} + {{ - 3\left( {x + 3} \right)} \over {{{\left( {x - 3} \right)}^2}\left( {x + 3} \right)}} \) \(\displaystyle+ {{ - x\left( {x - 3} \right)} \over {{{\left( {x - 3} \right)}^2}\left( {x + 3} \right)}}\)
\(\displaystyle = {{18 - 3x - 9 - {x^2} + 3x} \over {{{\left( {x - 3} \right)}^2}\left( {x + 3} \right)}} \)\(\displaystyle = {{9 - {x^2}} \over {\left( {3 - {x}} \right)^2\left( {x + 3} \right)}} \)
\(\displaystyle = {{\left( {3 - x} \right)\left( {3 + x} \right)} \over {\left( {3 - {x}} \right)^2\left( {x + 3} \right)}} = {1 \over {3 - x}} \)
