Đề bài

Thực hiện các phép tính sau: a) \(\dfrac{{x + 2}}{{x - 1}} - \dfrac{{x - 3}}{{x - 1}} + \dfrac{{x - 4}}{{1 - x}}\) b) \(\dfrac{1}{{x + 5}} - \dfrac{1}{{x - 5}} + \dfrac{{2x}}{{{x^2} - 25}}\) c) \(x + \dfrac{{2{y^2}}}{{x + y}} - y\)

Lời giải chi tiết

a) ĐKXĐ: \(x \ne 1\) \(\dfrac{{x + 2}}{{x - 1}} - \dfrac{{x - 3}}{{x - 1}} + \dfrac{{x - 4}}{{1 - x}}\) \( = \dfrac{{x + 2}}{{x - 1}} - \dfrac{{x - 3}}{{x - 1}} - \dfrac{{x - 4}}{{x - 1}} \) \(= \dfrac{{x + 2 - x + 3 - x + 4}}{{x - 1}} \) \(= \dfrac{{9 - x}}{{x - 1}}\) b) ĐKXĐ: \(x \ne  \pm 5\) \(\dfrac{1}{{x + 5}} - \dfrac{1}{{x - 5}} + \dfrac{{2x}}{{{x^2} - 25}}\) \( = \dfrac{{\left( {x - 5} \right)}}{{\left( {x + 5} \right)\left( {x - 5} \right)}} - \dfrac{{\left( {x + 5} \right)}}{{\left( {x + 5} \right)\left( {x - 5} \right)}} + \dfrac{{2x}}{{\left( {x + 5} \right)\left( {x - 5} \right)}}\) \(= \dfrac{{x - 5 - x - 5 + 2x}}{{\left( {x + 5} \right)\left( {x - 5} \right)}} \) \(= \dfrac{{2x - 10}}{{\left( {x + 5} \right)\left( {x - 5} \right)}}\) \( = \dfrac{{2\left( {x - 5} \right)}}{{\left( {x + 5} \right)\left( {x - 5} \right)}} \) \(= \dfrac{2}{{x + 5}}\)   c) ĐKXĐ: \(x \ne  - y\) \(x + \dfrac{{2{y^2}}}{{x + y}} - y\) \( = \dfrac{{x\left( {x + y} \right)}}{{x + y}} + \dfrac{{2{y^2}}}{{x + y}} - \dfrac{{y\left( {x + y} \right)}}{{x + y}} \) \(= \dfrac{{{x^2} + xy + 2{y^2} - xy - {y^2}}}{{x + y}} \) \(= \dfrac{{{x^2} + {y^2}}}{{x + y}}\)