lời giải

a)

\(\left(x+1\right)\left(2x-1\right)+x\le2x^2+3\)

\(\Leftrightarrow2x^2+x-1+x\le2x^2+3\)

\(\Leftrightarrow2x\le4\Rightarrow x\le2\)

\(\)b) \(\left(x+1\right)\left(x+2\right)\left(x+3\right)-x>x^3+6x^2-5\)

\(\left(x^2+3x+2\right)\left(x+3\right)-x>x^3+6x^2-5\)

\(x^3+3x^2+3x^2+9x+2x+6-x>x^3+6x^2-5\)

\(10x+6>-5\Rightarrow x>-\dfrac{11}{10}\)

c)Đkxđ: x≥0
x+√x>(2√x+3)(√x−1)
⇔x+√x>2x+√x−3
⇔x−3>0
⇔x>3. (tmđk).
 

a) ĐK: \(\orbr{\begin{cases}x\ge3+\sqrt{3}\\x\le3-\sqrt{3}\end{cases}}\)

pt \(\Leftrightarrow\)\(x^2-6x+9-4\sqrt{x^2-6x+6}=0\)

\(\Leftrightarrow\)\(a^2-4a+3=0\)\(\left(a=\sqrt{x^2-6x+6}\ge0\right)\)

\(\Leftrightarrow\)\(\orbr{\begin{cases}\sqrt{x^2-6x+6}=1\\\sqrt{x^2-6x+6}=3\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=1hoacx=5\\x=3\pm2\sqrt{3}\end{cases}}\left(nhan\right)\)

b) ĐK.. 

pt \(\Leftrightarrow\)\(\frac{\left(x-2\right)^2}{\left(x-1\right)^2}+2\left|\frac{x-2}{x-1}\right|-3=0\)

\(\Leftrightarrow\)\(\orbr{\begin{cases}\left|\frac{x-2}{x-1}\right|=-3\left(loai\right)\\\left|\frac{x-2}{x-1}\right|=1\end{cases}}\Leftrightarrow x=\frac{3}{2}\left(nhan\right)\)

Lời giải

a) \(\sqrt{\left(x-4\right)^2\left(x+1\right)}>0\Leftrightarrow\left\{{}\begin{matrix}x\ne4\\x+1>0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x\ne4\\x>-1\end{matrix}\right.\)

b) \(\sqrt{\left(x+2\right)^2\left(x-3\right)}>0\Rightarrow\left\{{}\begin{matrix}x\ne-2\\x-3>0\end{matrix}\right.\) \(\Rightarrow x>3\)

\(\sqrt{\left(x+1\right)\left(4-x\right)}>x-2\)   (1)

\(\Leftrightarrow\)   \(\begin{cases}x-2<0\\\left(x+1\right)\left(4-x\right)\ge0\end{cases}\)   hoặc \(\begin{cases}x-2\ge0\\\left(x+1\right)\left(4-x\right)>\left(x-2\right)^2\end{cases}\)

\(\Leftrightarrow\) \(\begin{cases}x<2\\-1\le x\le4\end{cases}\)  hoặc \(\begin{cases}2\le x\\2x^2-7x<0\end{cases}\)

\(\Leftrightarrow\)  \(\begin{cases}x<2\\-1\le x\le4\end{cases}\) hoặc (2\(\le\) x; 0 < x < \(\frac{7}{2}\)

\(\Leftrightarrow\)  \(-1\le x<2\)  hoặc \(2\le x<\frac{7}{2}\)

\(\Leftrightarrow\) \(-1\le x<\frac{7}{2}\)

Vậy bất phương trình đã cho có nghiệm  \(-1\le x<\frac{7}{2}\)