a) \(\dfrac{{x + 1}}{{x - 1}} + 2 > \dfrac{{x - 1}}{x}\) \( \Leftrightarrow \dfrac{{3x - 1}}{{x - 1}} > \dfrac{{x - 1}}{x}\) \( \Leftrightarrow \dfrac{{3{x^2} - x - {{(x - 1)}^2}}}{{x(x - 1)}} > 0\) \( \Leftrightarrow \dfrac{{2{x^2} + x - 1}}{{x(x - 1)}} > 0\)
\( \Leftrightarrow x < - 1\)hoặc \(0 < x < \dfrac{1}{2}.\)hoặc \(x > 1\).
b) \(\dfrac{1}{{x + 1}} + \dfrac{2}{{x + 3}} + \dfrac{3}{{x + 2}} < 0\) \( \Leftrightarrow \dfrac{{x + 3 + 2x + 2}}{{(x + 1)(x + 3)}} < \dfrac{3}{{x + 2}}\)
\( \Leftrightarrow \dfrac{{(3x + 5)(x + 2) - 3(x + 1) + (x + 3)}}{{(x + 1)(x + 2)(x + 3)}} < 0.\)
\( \Leftrightarrow \dfrac{{1 - x}}{{(x + 1)(x + 2)(x + 3)}} < 0\)
\( \Leftrightarrow x < - 3\)hoặc \( - 2 < x < - 1\) hoặc \(x > 1\).
Đáp số: \(x < - 3\)hoặc \( - 2 < x < - 1\) hoặc \(x > 1\).
