\(C=\left(\dfrac{2x-3}{4x^2-12x+5}+\dfrac{3}{1-2x}-\dfrac{2x-8}{2x^2-13x+20}\right):\dfrac{21+2x-8x^2}{4x^2-3+4x}+1\)
a) Rút gọn C
b) Tính C khi \(\left|x\right|=1\)
c) Tìm x thuộc Z để C thuộc Z
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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/ \(P>0\Rightarrow\dfrac{2}{2x-5}+1>0\Rightarrow\dfrac{2x-3}{2x-5}>0\)
\(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}2x-3>0\\2x-5>0\end{matrix}\right.\\\left\{{}\begin{matrix}2x-3< 0\\2x-5< 0\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x< \dfrac{3}{2}\\x< \dfrac{5}{2}\end{matrix}\right.\\\left\{{}\begin{matrix}x>\dfrac{3}{2}\\x>\dfrac{5}{2}\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x< \dfrac{3}{2}\\x>\dfrac{5}{2}\end{matrix}\right.\)
a. ĐKXĐ : \(x\ne\frac{1}{2};\frac{5}{2};4;-\frac{3}{2};\frac{1\pm\sqrt{43}}{2}\)
\(A=\left(\frac{2x-3}{4x^2-12x+5}+\frac{3x-8}{13x-2x^2-20}-\frac{3}{2x-1}\right):\frac{21+2x-2x^2}{4x^2+4x-3}+\)
\(=\left(\frac{2x-3}{\left(2x-1\right)\left(2x-5\right)}-\frac{3x-8}{\left(2x-5\right)\left(x-4\right)}-\frac{3}{2x-1}\right).\frac{\left(2x-1\right)\left(2x+3\right)}{21+2x-2x^2}+1\)
\(=\frac{\left(2x-3\right)\left(x-4\right)-\left(3x-8\right)\left(2x-1\right)-3\left(2x-5\right)\left(x-4\right)}{\left(2x-1\right)\left(2x-5\right)\left(x-4\right)}.\frac{\left(2x-1\right)\left(2x+3\right)}{21+2x-2x^2}+1\)
\(=\frac{-10x^2+47x-56}{\left(2x-5\right)\left(x-4\right)}.\frac{2x+3}{-2x^2+2x+21}+1\) số to wa
`a,ĐKXĐ:x-4 ne 0,2x+2 ne 0`
`<=>x ne 4,x me -1`
`b,ĐKXĐ:4x^2-25 ne 0`
`<=>(2x-5)(2x+5) ne 0`
`<=>x ne +-5/2`
`c,ĐKXĐ:8x^3+27 ne 0`
`<=>8x^3 ne -27`
`<=>2x ne -3`
`<=>x ne -3/2`
`d,2x+2 ne 0,4y^2-9 ne 0`
`<=>2x ne -2,(2y-3)(2y+3) ne 0`
`<=>x ne -1,y ne +-3/2`
b) ĐKXĐ: \(x\notin\left\{\dfrac{5}{2};-\dfrac{5}{2}\right\}\)
c) ĐKXĐ: \(x\ne-\dfrac{3}{2}\)
d) ĐKXĐ: \(\left\{{}\begin{matrix}x\ne-1\\y\notin\left\{\dfrac{3}{2};-\dfrac{3}{2}\right\}\end{matrix}\right.\)
a: \(=\left(\dfrac{2x-3}{\left(2x-5\right)\left(2x-1\right)}-\dfrac{2x-8}{\left(2x-5\right)\left(x-4\right)}-\dfrac{3}{2x-1}\right)\cdot\dfrac{4x^2+4x-3}{-8x^2+2x+21}+1\)
\(=\dfrac{2x-3-2\left(2x-1\right)-3\left(x-4\right)}{\left(2x-1\right)\left(2x-5\right)}\cdot\dfrac{\left(2x+3\right)\left(2x-1\right)}{\left(4x-7\right)\cdot\left(-2x-3\right)}+1\)
\(=\dfrac{2x-3-4x+2-3x+12}{1}\cdot\dfrac{-1}{4x-7}+1\)
\(=\dfrac{5x-11}{4x-7}+1=\dfrac{9x-4}{4x-7}\)
b: |x|=1/2
=>x=1/2(loại) hoặc x=-1/2(nhận)
Khi x=-1/2 thì \(P=\dfrac{\dfrac{-9}{2}-4}{-2-7}=-\dfrac{17}{2}:\left(-9\right)=\dfrac{17}{18}\)
c: Để P là số nguyên thì \(36x-16⋮4x-7\)
\(\Leftrightarrow4x-7\in\left\{1;-1;47;-47\right\}\)
hay \(x\in\left\{2;\dfrac{3}{2};\dfrac{27}{2};-10\right\}\)
\(=\left[\dfrac{2x-3}{\left(2x-5\right)\left(2x-1\right)}-\dfrac{3}{2x-1}-\dfrac{2\left(x-4\right)}{\left(x-4\right)\left(2x-5\right)}\right].\dfrac{2x\left(2x+3\right)-\left(2x+3\right)}{-2x\left(4x-7\right)-3\left(4x-7\right)}+1\)
\(=\left[\dfrac{2x-3-6x+15-4x+2}{\left(2x-5\right)}\right].\dfrac{2\left(x+\dfrac{3}{2}\right)}{\left(-2x-3\right)\left(4x-7\right)}+1\)
\(=\dfrac{-2\left(4x-7\right)}{2x-5}.\dfrac{2\left(x+\dfrac{3}{2}\right)}{\left(-2x-3\right)\left(4x-7\right)}+1\)
\(=\dfrac{1}{2x-5}.2+1\)
\(=\dfrac{2+2x-5}{2x-5}\)
\(=\dfrac{-3+2x}{2x-5}\)