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\(\Leftrightarrow\dfrac{3}{\left(x+1\right)\left(x+4\right)}+\dfrac{3}{\left(x+4\right)\left(x+7\right)}+\dfrac{3}{\left(x+7\right)\left(x+10\right)}+\dfrac{3}{\left(x+10\right)\left(x+13\right)}=\dfrac{12}{13}\)
\(\Leftrightarrow\dfrac{1}{x+1}-\dfrac{1}{x+13}=\dfrac{12}{13}\)
\(\Leftrightarrow12\left(x+1\right)\left(x+13\right)=13\left(x+13\right)-13\left(x+1\right)=156\)
\(\Leftrightarrow\left(x+1\right)\left(x+13\right)=13\)
\(\Leftrightarrow x^2+14x=0\)
=>x=0 hoặc x=-14
ĐKXĐ: \(x\ne0\)
Đặt \(\dfrac{x}{2}-\dfrac{3}{x}=a\Rightarrow\dfrac{x^2}{4}+\dfrac{9}{x^2}-3=a^2\Rightarrow\dfrac{x^2}{4}+\dfrac{9}{x^2}=a^2+3\Rightarrow\dfrac{x^2}{2}+\dfrac{18}{x^2}=2a^2+6\)
Pt đã cho trở thành:
\(2a^2+6=13a\Leftrightarrow2a^2-13a+6=0\Rightarrow\left[{}\begin{matrix}a=6\\a=\dfrac{1}{2}\end{matrix}\right.\)
TH1: \(a=6\Rightarrow\dfrac{x}{2}-\dfrac{3}{x}=6\Leftrightarrow\dfrac{x^2-6}{2x}=6\Leftrightarrow x^2-12x-6=0\)
\(\Rightarrow\left[{}\begin{matrix}x=6-\sqrt{42}\\x=6+\sqrt{42}\end{matrix}\right.\)
TH2: \(a=\dfrac{1}{2}\Rightarrow\dfrac{x}{2}-\dfrac{3}{x}=\dfrac{1}{2}\Leftrightarrow\dfrac{x^2-6}{2x}=\dfrac{1}{2}\Leftrightarrow x^2-x-6=0\)
\(\Rightarrow\left[{}\begin{matrix}x=3\\x=-2\end{matrix}\right.\)
Vậy pt đã cho có 4 nghiệm
a: \(\Leftrightarrow\dfrac{7x+10}{x+1}\left(x^2-x-2-2x^2+3x+5\right)=0\)
\(\Leftrightarrow\left(7x+10\right)\left(-x^2+2x+3\right)=0\)
\(\Leftrightarrow\left(7x+10\right)\cdot\left(x^2-2x-3\right)=0\)
=>(7x+10)(x-3)=0
=>x=3 hoặc x=-10/7
b: \(\Leftrightarrow\dfrac{13}{\left(2x+7\right)\left(x-3\right)}+\dfrac{1}{2x+7}-\dfrac{6}{\left(x-3\right)\left(x+3\right)}=0\)
\(\Leftrightarrow13\left(x+3\right)+x^2-9-12x-42=0\)
\(\Leftrightarrow x^2-12x-51+13x+39=0\)
\(\Leftrightarrow x^2+x-12=0\)
=>(x+4)(x-3)=0
=>x=-4
a, \(\dfrac{4x+13}{5x\left(x-7\right)}-\dfrac{x-48}{5x\left(7-x\right)}\)
\(=\dfrac{4x+13}{5x\left(x-7\right)}+\dfrac{x-48}{5x\left(x-7\right)}\)
\(=\dfrac{4x+13+x-48}{5x\left(x-7\right)}\)
\(=\dfrac{5x-35}{5x\left(x-7\right)}\)
\(=\dfrac{5\left(x-7\right)}{5x\left(x-7\right)}=\dfrac{1}{x}\)
b, \(\dfrac{1}{x-5x^2}-\dfrac{25x-15}{25x^2-1}\)
\(=\dfrac{1}{x\left(1-5x\right)}+\dfrac{25x-15}{\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x}{x\left(x-5x\right)\left(1+5x\right)}+\dfrac{x\left(25x-15\right)}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x+25x^2-15x}{x\left(1-5x\right)\left(1+5x\right)}\)\(=\dfrac{25x^2-10x+1}{x\left(1-5x\right)\left(1+5x\right)}=\dfrac{\left(5x-1\right)^2}{x.\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{\left(5x-1\right)^2}{-x\left(5x-1\right)\left(1+5x\right)}\) \(=\dfrac{-\left(5x-1\right)}{x\left(1+5x\right)}\)
a: =>-12x>12
hay x<-1
b: =>7(3x-1)-252>=21x+3(6x+1)
=>21x-7-252>=21x+18x+3
=>18x+3<=-259
=>18x<=-262
hay x<=-131/9
c: =>3(3x+5)-24x<=48+4(x+8)
=>9x+15-24x<=48+4x+32=4x+80
=>-15x+24<=4x+80
=>-19x<=56
hay x>=-56/19
b: \(\Leftrightarrow\dfrac{7x+10}{x+1}\left(x^2-x-2-2x^2+3x+5\right)=0\)
\(\Leftrightarrow\left(7x+10\right)\left(-x^2+2x+3\right)=0\)
\(\Leftrightarrow\left(7x+10\right)\left(x^2-2x-3\right)=0\)
=>(7x+10)(x-3)=0
hay \(x\in\left\{-\dfrac{10}{7};3\right\}\)
d: \(\Leftrightarrow\dfrac{13}{2x^2+7x-6x-21}+\dfrac{1}{2x+7}-\dfrac{6}{\left(x-3\right)\left(x+3\right)}=0\)
\(\Leftrightarrow\dfrac{13}{\left(2x+7\right)\left(x-3\right)}+\dfrac{1}{\left(2x+7\right)}-\dfrac{6}{\left(x-3\right)\left(x+3\right)}=0\)
\(\Leftrightarrow26x+91+x^2-9-12x-14=0\)
\(\Leftrightarrow x^2+14x+68=0\)
hay \(x\in\varnothing\)
