ĐKXĐ: \(x\ne\left\{\dfrac{-3}{2};\dfrac{1}{2};\dfrac{7}{4};\dfrac{5}{2};4;\right\}\)
\(P=\left(\dfrac{2x-3}{\left(2x-1\right)\left(2x-5\right)}-\dfrac{3}{2x-1}-\dfrac{2\left(x-4\right)}{\left(2x-5\right)\left(x-4\right)}\right)\div\dfrac{\left(7-4x\right)\left(2x+3\right)}{\left(2x-1\right)\left(2x+3\right)}+1\)
\(P=\left(\dfrac{2x-3-3\left(2x-5\right)-2\left(2x-1\right)}{\left(2x-1\right)\left(2x-5\right)}\right)\dfrac{2x-1}{7-4x}+1\)
\(P=\dfrac{-8x+14}{\left(2x-5\right)\left(7-4x\right)}+1=\dfrac{2}{2x-5}+1\)
b/ \(\left|x\right|=\dfrac{1}{2}\Rightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}\\x=\dfrac{-1}{2}\end{matrix}\right.\)
Với \(x=\dfrac{1}{2}\Rightarrow P=\dfrac{2}{2.\dfrac{1}{2}-5}+1=\dfrac{1}{2}\)
Với \(x=\dfrac{-1}{2}\Rightarrow P=\dfrac{2}{2.\left(\dfrac{-1}{2}\right)-5}+1=\dfrac{2}{3}\)
c/ Để P nguyên \(\Rightarrow\dfrac{2}{2x-5}\) nguyên \(\Rightarrow2⋮\left(2x-5\right)\Rightarrow2x-5=Ư\left(2\right)=\left\{-2;-1;1;2\right\}\)
\(2x-5=-2\Rightarrow x=\dfrac{3}{2}\left(l\right)\)
\(2x-5=-1\Rightarrow x=2\)
\(2x-5=1\Rightarrow x=3\)
\(2x-5=2\Rightarrow x=\dfrac{7}{2}\left(l\right)\)
Vậy \(x=\left\{2;3\right\}\) thì P nguyên
d/ \(P>0\Rightarrow\dfrac{2}{2x-5}+1>0\Rightarrow\dfrac{2x-3}{2x-5}>0\)
\(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}2x-3>0\\2x-5>0\end{matrix}\right.\\\left\{{}\begin{matrix}2x-3< 0\\2x-5< 0\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x< \dfrac{3}{2}\\x< \dfrac{5}{2}\end{matrix}\right.\\\left\{{}\begin{matrix}x>\dfrac{3}{2}\\x>\dfrac{5}{2}\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x< \dfrac{3}{2}\\x>\dfrac{5}{2}\end{matrix}\right.\)
