Đề bài
Thực hiện phép tính:
a) (frac{1}{{x – 2}} – frac{1}{{x + 1}})
b) (frac{{12}}{{{x^2} – 9}} – frac{2}{{x – 3}})
c) (frac{1}{{xy – {x^2}}} – frac{1}{{{y^2} – xy}})
d) (frac{{2{rm{x}}}}{{{x^2} – 1}} – frac{3}{{2 + 2{rm{x}}}} + frac{1}{{2 – 2{rm{x}}}})
Lời giải chi tiết
a)
(frac{1}{{x – 2}} – frac{1}{{x + 1}} = frac{{x + 1}}{{left( {x – 2} right)left( {x + 1} right)}} – frac{{x – 2}}{{left( {x – 2} right)left( {x + 1} right)}})
(= frac{{x + 1 – x + 2}}{{left( {x – 2} right)left( {x + 1} right)}} = frac{3}{{left( {x – 2} right)left( {x + 1} right)}})
b)
(begin{array}{l}frac{{12}}{{{x^2} – 9}} – frac{2}{{x – 3}} = frac{{12}}{{left( {x – 3} right)left( {x + 3} right)}} – frac{2}{{x – 3}} = frac{{12}}{{left( {x – 3} right)left( {x + 3} right)}} – frac{{2left( {x + 3} right)}}{{left( {x – 3} right)left( {x + 3} right)}} = frac{{12 – 2{rm{x}} – 6}}{{left( {x – 3} right)left( {x + 3} right)}} = frac{{6 – 2{rm{x}}}}{{left( {x – 3} right)left( {x + 3} right)}} = frac{{ – 2left( {x – 3} right)}}{{left( {x – 3} right)left( {x + 3} right)}} = frac{{ – 2}}{{x + 3}}end{array})
c)
(begin{array}{l}frac{1}{{xy – {x^2}}} – frac{1}{{{y^2} – xy}} = frac{1}{{xleft( {y – x} right)}} – frac{1}{{yleft( {y – x} right)}} = frac{y}{{xyleft( {y – x} right)}} – frac{x}{{xyleft( {y – x} right)}} = frac{{y – x}}{{xyleft( {y – x} right)}} = frac{1}{{xy}}end{array})
d)
(begin{array}{l}frac{{2{rm{x}}}}{{{x^2} – 1}} – frac{3}{{2 + 2{rm{x}}}} + frac{1}{{2 – 2{rm{x}}}} = frac{{2{rm{x}}}}{{left( {x – 1} right)left( {x + 1} right)}} – frac{3}{{2left( {x + 1} right)}} – frac{1}{{2{rm{x}} – 2}} = frac{{2{rm{x}}}}{{left( {x – 1} right)left( {x + 1} right)}} – frac{{3left( {x – 1} right)}}{{2left( {x + 1} right)left( {x – 1} right)}} – frac{1}{{2left( {x – 1} right)}} = frac{{{rm{4x}}}}{{2left( {x – 1} right)left( {x + 1} right)}} – frac{{3left( {x – 1} right)}}{{2left( {x + 1} right)left( {x – 1} right)}} – frac{{1left( {x + 1} right)}}{{2left( {x – 1} right)left( {x + 1} right)}} = frac{{{rm{4x}} – 3{rm{x}} + 3 – x – 1}}{{2left( {x – 1} right)left( {x + 1} right)}} = frac{2}{{2left( {x – 1} right)left( {x + 1} right)}} = frac{1}{{left( {x – 1} right)left( {x + 1} right)}}end{array})
