a) \(|3x+5|=4x-1\) (1)
TH1: \(3x+5\ge 0\Leftrightarrow x\ge\dfrac{-5}{3}\)
(1) \(\Rightarrow 3x+5=4x-1\)
\(\Leftrightarrow x=6\) (tm).
TH2: \(3x+5< 0\Leftrightarrow x<\dfrac{-5}{3}\)
(1) \(\Rightarrow -(3x+5)=4x-1\)
\(\Leftrightarrow x=\dfrac{-4}{7}(>\dfrac{-5}{3})\) (loại).
b) \(\sqrt{10-x}=x+2\) (2)
ĐK: \(10-x\ge0\Leftrightarrow x\le 10\)
(2) \(\Rightarrow 10-x=(x+2)^2\)
\(\Leftrightarrow 10-x=x^2+4x+4\)
\(\Leftrightarrow x^2+5x-6=0\)
\(\Leftrightarrow x^2-x+6x-6=0\)
\(\Leftrightarrow x(x-1)+6(x-1)=0\)
\(\Leftrightarrow (x-1)(x+6)=0\)
\(\Leftrightarrow \left\{ \begin{array}{l} x-1=0 \\ x+6=0 \end{array} \right .\Leftrightarrow\left\{ \begin{array}{l} x=1 (tm)\\ x=-6 (tm).\end{array} \right .\)
3) \(\dfrac{x^2+x-6}{-3x+11}>0\)
\(\Leftrightarrow \dfrac{x^2+3x-2x-6}{-3x+11}>0\)
\(\Leftrightarrow \dfrac{x(x+3)-2(x+3)}{-3x+11}>0\)
\(\Leftrightarrow \dfrac{(x+3)(x-2)}{-3x+11}>0\)
\(\Leftrightarrow \left[ \begin{array}{l} -3\le x\le2 \\ x>\dfrac{11}{3} \end{array} \right .\)
d) \(\dfrac{1}{x+2}+\dfrac{2}{x-1}\le\dfrac{3}{x+1}\)
ĐK: \(x\ne \{-2;\pm1\}\)
Bpt \(\Leftrightarrow \dfrac{1}{x+2}+\dfrac{2}{x-1}-\dfrac{3}{x+1}\le0\)
\(\Leftrightarrow \dfrac{(x-1)(x+1)+2(x+2)(x+1)-3(x+2)(x-1)}{(x+2)(x-1)(x+1)}\le0\)
\(\Leftrightarrow \dfrac{x^2-1+2(x^2+x+2x+2)-3(x^2-x+2x-2)}{(x+2)(x-1)(x+1)}\le 0\)
\(\Leftrightarrow \dfrac{3x+9}{(x+2)(x-1)(x+1)}\le 0\)
\(\Leftrightarrow \dfrac{3(x+1)}{(x+2)(x-1)(x+1)}\le0\)
\(\Leftrightarrow \dfrac{3}{(x+2)(x-1)}\le0\)
\(\Leftrightarrow \left[ \begin{array}{l} x<-2 \\ x>1 \end{array} \right .\)
