Giải hệ pt
a) \(\left\{{}\begin{matrix}x+\dfrac{y}{\sqrt{1+x^2}+x}+y^2=0\\\dfrac{x^2}{y^2}+2\sqrt{x^2+1}+y^2=3\end{matrix}\right.\)
b) \(\left\{{}\begin{matrix}\sqrt{x^2+4}+\sqrt{y^2+2y-4}=4\\\sqrt{x^2+9}+y=5\end{matrix}\right.\)
Giải hệ
a) \(\left\{{}\begin{matrix}x^2+y^2-2y-6+2\sqrt{2y+3}=0\\\left(x-y\right)\left(x^2+xy+y^2+3\right)=3\left(x^2+y^2\right)+2\end{matrix}\right.\)
b) \(\left\{{}\begin{matrix}x^2y+2y+x=4xy\\\dfrac{1}{x^2}+\dfrac{1}{xy}+\dfrac{x}{y}=3\end{matrix}\right.\)
Giải hệ bất phuơng trình:
\(\left\{{}\begin{matrix}\dfrac{1}{\sqrt{x}}+\dfrac{y}{x}=\dfrac{2\sqrt{x}}{y}+2\\y\left(\sqrt{x^2+1}-1\right)=\sqrt{3x^2+3}\end{matrix}\right.\)
Tìm GTNN
a) \(y=\sqrt{x^3+2\left(1+\sqrt{x^3+1}\right)}+\sqrt{x^3+2\left(1-\sqrt{x^3+1}\right)}\)
b) \(f\left(x\right)=\dfrac{x}{2}+\dfrac{2}{x-1}\) với x>1
c) \(y=\dfrac{x-2017}{\sqrt{x-2018}}\)
1. Cho \(x,y,z\) là 3 số thực dương thõa mản xyz = 1. C/m BĐT
\(\dfrac{1}{\left(2x+y+z\right)^2}+\dfrac{1}{\left(2x+y+z\right)^2}+\dfrac{1}{\left(2x+y+z\right)^2}\le\dfrac{3}{16}\)
2. Cho x,y,z không âm và thõa mản \(x^2+y^2+z^2=1\). C/m BĐT
\(\left(x^2y+y^2z+z^2x\right)\left(\dfrac{1}{\sqrt{x^2+1}}+\dfrac{1}{\sqrt{y^2+1}}+\dfrac{1}{\sqrt{z^2+1}}\right)\le\dfrac{3}{2}\)
giúp mình giải bpt vs
\(\dfrac{\left|2x-1\right|-x}{2x}>1;\dfrac{2-\left|x-2\right|}{x^2-1}\ge0;\dfrac{\sqrt{x+4}-2}{4-9x^2}\le0;\dfrac{x^2-2x-3}{\sqrt[3]{3x-1}+\sqrt[3]{4-5x}}\ge0;\)\(3x^2-10x+3\ge0;\left(\sqrt{2}-x\right)\left(x^2-2\right)\left(2x-4\right)< 0;\dfrac{1}{x+9}-\dfrac{1}{x}>\dfrac{1}{2};\dfrac{2}{1-2x}\le\dfrac{3}{x+1}\)
Giải các bất phương trình, hệ phương trình
a) \(\dfrac{x^2\left(3x-2\right)\left(x^2-1\right)}{\left(-x^2+2x-3\right)\left(2-x\right)^2}\ge0\)
b) \(\dfrac{x-5}{x-1}>2\)
c) \(2x-\sqrt{x^2-5x-14}< 1\)
d) \(x+\sqrt{x^2-4x-5}< 4\)
e) \(\left\{{}\begin{matrix}\left(4-x\right)\left(x^2-2x-3\right)< 0\\x^2\ge\left(x^2-x-3\right)^2\end{matrix}\right.\)
\(\sqrt{x+y\left(x-1\right)}+\sqrt{x}=y+\sqrt{y}\\ \left(x-1\right)^2+y\sqrt{\left(x-\dfrac{1}{y}\right)^3}=2\)
giải hệ phương trình
a) \(\left\{{}\begin{matrix}\sqrt{2x^2+2y^2}+\sqrt{\frac{4}{3}\left(x^2+xy+y^2\right)}=2\left(x+y\right)\\\sqrt{3x+1}+\sqrt{5x+4}=3xy-y+3\end{matrix}\right.\)
b) \(\left\{{}\begin{matrix}\sqrt{5x^2+2xy+2y^2}+\sqrt{2x^2+2xy+5y^2}=3\left(x+y\right)\\\sqrt{x+2y+1}+2\sqrt[3]{12x+7y+8}=2xy+x+5\end{matrix}\right.\)
c)\(\left\{{}\begin{matrix}x^2+xy+x+3=0\\\left(x+1\right)^2+3\left(y+1\right)+2\left(xy-\sqrt{x^2y+2y}\right)=0\end{matrix}\right.\)
