1: \(\left|x-3,5\right|>=0\forall x\)

\(\left|4,5-y\right|>=0\forall y\)

Do đó: \(\left|x-3,5\right|+\left|4.5-y\right|>=0\forall x,y\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x-3,5=0\\4,5-y=0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=3,5\\y=4,5\end{matrix}\right.\)

2: \(\left\{{}\begin{matrix}\left|x+\dfrac{2}{3}\right|>=0\forall x\\\left|y-\dfrac{3}{4}\right|>=0\forall y\\\left|z-5\right|>=0\forall z\end{matrix}\right.\)

Do đó: \(\left|x+\dfrac{2}{3}\right|+\left|y-\dfrac{3}{4}\right|+\left|z-5\right|>=0\forall x,y,z\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x+\dfrac{2}{3}=0\\y-\dfrac{3}{4}=0\\z-5=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{2}{3}\\y=\dfrac{3}{4}\\z=5\end{matrix}\right.\)

3: \(\left|x-2\right|+\left|3-x\right|=0\)

=>|x-2|+|x-3|=0(1)

TH1: x<2

Phương trình (1) sẽ trở thành 2-x+3-x=0

=>5-2x=0

=>2x=5

=>x=2,5(loại)

TH2: 2<=x<3

Phương trình (1) sẽ trở thành x-2+3-x=0

=>1=0(loại)

TH3: x>=3

Phương trình (1) sẽ trở thành x-2+x-3=0

=>2x=5

=>x=2,5(loại)

Vậy: Phương trình vô nghiệm

4: \(\left\{{}\begin{matrix}\left|x-\dfrac{2}{3}\right|>=0\forall x\\\left|x+y+\dfrac{3}{4}\right|>=0\forall x,y\\\left|y-z-\dfrac{5}{6}\right|>=0\forall y,z\end{matrix}\right.\)

Do đó: \(\left|x-\dfrac{2}{3}\right|+\left|x+y+\dfrac{3}{4}\right|+\left|y-z-\dfrac{5}{6}\right|>=0\forall x,y,z\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x-\dfrac{2}{3}=0\\x+y+\dfrac{3}{4}=0\\y-z-\dfrac{5}{6}=0\end{matrix}\right.\)

=>\(\begin{matrix}x=\dfrac{2}{3}\\y=-x-\dfrac{3}{4}=-\dfrac{2}{3}-\dfrac{3}{4}=\dfrac{-17}{12}\\z=y-\dfrac{5}{6}=-\dfrac{17}{12}-\dfrac{5}{6}=-\dfrac{27}{12}=-\dfrac{9}{4}\end{matrix}\)

5: \(\left\{{}\begin{matrix}\left|x-\dfrac{2}{3}\right|>=0\forall x\\\left|xy-\dfrac{5}{8}\right|>=0\forall x,y\\\left|yz+\dfrac{3}{4}\right|>=0\forall y,z\end{matrix}\right.\)

Do đó: \(\left|x-\dfrac{2}{3}\right|+\left|xy-\dfrac{5}{8}\right|+\left|yz+\dfrac{3}{4}\right|>=0\forall x,y,z\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x-\dfrac{2}{3}=0\\xy-\dfrac{5}{8}=0\\yz+\dfrac{3}{4}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{2}{3}\\xy=\dfrac{5}{8}\\yz=-\dfrac{3}{4}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{2}{3}\\y=\dfrac{5}{8}:x=\dfrac{5}{8}:\dfrac{2}{3}=\dfrac{5}{8}\cdot\dfrac{3}{2}=\dfrac{15}{16}\\z=-\dfrac{3}{4}:\dfrac{15}{16}=-\dfrac{3}{4}\cdot\dfrac{16}{15}=\dfrac{-48}{60}=-\dfrac{4}{5}\end{matrix}\right.\)

6: \(\left\{{}\begin{matrix}\left|xy+\dfrac{2}{3}\right|>=0\forall x,y\\\left|yz-\dfrac{8}{9}\right|>=0\forall y,z\\\left|xz+\dfrac{3}{4}\right|>=0\forall x,z\end{matrix}\right.\)

Do đó: \(\left|xy+\dfrac{2}{3}\right|+\left|yz-\dfrac{8}{9}\right|+\left|xz+\dfrac{3}{4}\right|>=0\forall x,y,z\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}xy+\dfrac{2}{3}=0\\yz-\dfrac{8}{9}=0\\xz+\dfrac{3}{4}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}xy=-\dfrac{2}{3}\\yz=\dfrac{8}{9}\\xz=-\dfrac{3}{4}\end{matrix}\right.\)

=>\(\left(xyz\right)^2=-\dfrac{2}{3}\cdot\dfrac{8}{9}\cdot\dfrac{-3}{4}=\dfrac{1}{2}\cdot\dfrac{8}{9}=\dfrac{4}{9}\)

=>\(\left[{}\begin{matrix}xyz=\dfrac{2}{3}\\xyz=-\dfrac{2}{3}\end{matrix}\right.\)

TH1: xyz=2/3

=>\(\left\{{}\begin{matrix}z=\dfrac{xyz}{xy}=\dfrac{2}{3}:\dfrac{-2}{3}=-1\\x=\dfrac{xyz}{yz}=\dfrac{2}{3}:\dfrac{8}{9}=\dfrac{2}{3}\cdot\dfrac{9}{8}=\dfrac{18}{24}=\dfrac{3}{4}\\y=\dfrac{xyz}{xz}=\dfrac{2}{3}:\dfrac{-3}{4}=\dfrac{2}{3}\cdot\dfrac{4}{-3}=-\dfrac{8}{9}\end{matrix}\right.\)

TH2: xyz=-2/3

=>\(\left\{{}\begin{matrix}z=\dfrac{xyz}{xy}=-\dfrac{2}{3}:\dfrac{-2}{3}=1\\x=\dfrac{xyz}{yz}=-\dfrac{2}{3}:\dfrac{8}{9}=\dfrac{-2}{3}\cdot\dfrac{9}{8}=\dfrac{-18}{24}=\dfrac{-3}{4}\\y=\dfrac{xyz}{xz}=\dfrac{-2}{3}:\dfrac{-3}{4}=\dfrac{-2}{3}\cdot\dfrac{4}{-3}=\dfrac{8}{9}\end{matrix}\right.\)