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giải bài này theo cách này đc k ạ
\n\n\\(\\sqrt{a}\\le\\sqrt{b}\\Leftrightarrow\\left\\{{}\\begin{matrix}a\\ge0\\\\a< b\\end{matrix}\\right.\\)
\n\\(\\sqrt{a}\\le\\sqrt{b}\\Leftrightarrow\\left\\{{}\\begin{matrix}a\\ge0\\\\a\\le b\\end{matrix}\\right.\\)
\n\ne ghi lộn
\n![](https://rs.olm.vn/images/avt/0.png?1311)
Câu 6:
\(\hept{\begin{cases}\frac{x+3}{2x-3}-\frac{x}{2x-1}\le0\\\sqrt{x^2+3}+3< 1\end{cases}\Leftrightarrow\hept{\begin{cases}\frac{2x^2-x+6x-3-2x^2+3x}{\left(2x-3\right)\left(2x-1\right)}\le0\\x^2+3< \left(1-3x\right)^2\end{cases}}}\)
\(\Leftrightarrow\hept{\begin{cases}8x-3\le0\\x^2+3< 1-6x+9x^2\end{cases}\Leftrightarrow\hept{\begin{cases}8x-3\le0\\8x^2-6x-2< 0\end{cases}\Leftrightarrow}\hept{\begin{cases}x< \frac{3}{8}\\\frac{-1}{4}x< x< \frac{1}{4}\end{cases}\Rightarrow}S\left(\frac{-1}{4};\frac{3}{8}\right)}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a. ĐKXĐ: \(x\ge-1\)
\(y=\sqrt{x^3+1+2\sqrt{x^3+1}+1}+\sqrt{x^3+1-2\sqrt{x^3+1}+1}\)
\(=\sqrt{\left(\sqrt{x^3+1}+1\right)^2}+\sqrt{\left(\sqrt{x^3+1}-1\right)^2}\)
\(=\left|\sqrt{x^3+1}+1\right|+\left|1-\sqrt{x^3+1}\right|\ge\left|\sqrt{x^3+1}+1+1-\sqrt{x^3+1}\right|=2\)
b.
\(f\left(x\right)=\dfrac{x-1}{2}+\dfrac{2}{x-1}+\dfrac{1}{2}\ge2\sqrt{\dfrac{2\left(x-1\right)}{2\left(x-1\right)}}+\dfrac{1}{2}=\dfrac{5}{2}\)
c.
\(y=\dfrac{x-2018+1}{\sqrt{x-2018}}=\sqrt{x-2018}+\dfrac{1}{\sqrt{x-2018}}\ge2\sqrt{\dfrac{\sqrt{x-2018}}{\sqrt{x-2018}}}=2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a, ĐKXĐ:\(x\ne-3\)
\(x+1+\dfrac{2}{x+3}=\dfrac{x+5}{x+3}\\ \Leftrightarrow x+1=\dfrac{x+5}{x+3}-\dfrac{2}{x+3}\\ \Leftrightarrow x+1=\dfrac{x+3}{x+3}\\ \Leftrightarrow x+1=1\\ \Leftrightarrow x=0\left(tm\right)\)
b, ĐKXĐ:\(x>2\)
\(\dfrac{x^2-4x-2}{\sqrt{x-2}}=\sqrt{x-2}\\ \Leftrightarrow x^2-4x-2=x-2\\ \Leftrightarrow x^2-5x=0\\ \Leftrightarrow x\left(x-5\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\left(ktm\right)\\x=5\left(tm\right)\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
ĐKXĐ: \(x\ge1\)
Đặt \(\left\{{}\begin{matrix}\sqrt[]{x-1}=a\ge0\\\sqrt[3]{2-x}=b\end{matrix}\right.\) \(\Rightarrow a^2+b^3=1\)
Ta được hệ:
\(\left\{{}\begin{matrix}a+b=1\\a^2+b^3=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}b=1-a\\a^2+b^3=1\end{matrix}\right.\)
\(\Rightarrow a^2+\left(1-a\right)^3=1\)
\(\Leftrightarrow a^3-4a^2+3a=0\)
\(\Leftrightarrow a\left(a-1\right)\left(a-3\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}a=0\\a=1\\a=3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\sqrt[]{x-1}=0\\\sqrt[]{x-1}=1\\\sqrt[]{x-1}=3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=1\\x=2\\x=10\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
ĐKXĐ: \(-1\le x\le\dfrac{5}{2}\)
\(\Leftrightarrow\sqrt{3x+3}-3+1-\sqrt{5-2x}=x^3-3x^2-10x+24\)
\(\Leftrightarrow\dfrac{3\left(x-2\right)}{\sqrt{3x+3}+3}+\dfrac{2\left(x-2\right)}{1+\sqrt{5-2x}}=\left(x-2\right)\left(x-4\right)\left(x+3\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\\dfrac{3}{\sqrt{3x+3}+3}+\dfrac{2}{1+\sqrt{5-2x}}=\left(x-4\right)\left(x+3\right)\left(1\right)\end{matrix}\right.\)
Xét (1), ta có:
\(\dfrac{3}{\sqrt{3x+3}+3}+\dfrac{2}{1+\sqrt{5-2x}}>0\)
\(-1\le x\le\dfrac{5}{2}\Rightarrow\left\{{}\begin{matrix}x+3>0\\x-4< 0\end{matrix}\right.\) \(\Rightarrow\left(x+3\right)\left(x-4\right)< 0\)
\(\Rightarrow\left(1\right)\) vô nghiệm hay pt có nghiệm duy nhất \(x=2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a/ ĐKXĐ: \(\left\{{}\begin{matrix}x\le3\\x\ge5\end{matrix}\right.\) => bpt vô nghiệm
b/ ĐKXĐ: \(x>1\)
\(bpt\Leftrightarrow x-2< 2\Leftrightarrow x< 4\)
\(\Rightarrow1< x< 4\)
c/ \(\frac{x+2}{3}-2x-2>0\)
\(\Leftrightarrow\frac{x+2-6x-6}{3}>0\Leftrightarrow x+2-6x-6>0\Leftrightarrow x< -\frac{4}{5}\)
d/ \(bpt\Leftrightarrow\frac{3x+5}{2}-\frac{x+2}{3}-x-1\le0\)
\(\Leftrightarrow\frac{9x+15-2x-4-6x-6}{6}\le0\)
\(\Leftrightarrow x\le-5\)