Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
*)\\
\left| {x + 2} \right| = \dfrac{1}{2} \Leftrightarrow \left[ \begin{array}{l}
x + 2 = \dfrac{1}{2}\\
x + 2 =  - \dfrac{1}{2}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{1}{2} - 2\\
x =  - \dfrac{1}{2} - 2
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x =  - \dfrac{3}{2}\\
x =  - \dfrac{5}{2}
\end{array} \right.\\
*)\\
\left| {\dfrac{1}{4} - x} \right| - 2 = \dfrac{5}{2}\\
 \Leftrightarrow \left| {\dfrac{1}{4} - x} \right| = \dfrac{5}{2} + 2\\
 \Leftrightarrow \left| {\dfrac{1}{4} - x} \right| = \dfrac{9}{2}\\
 \Leftrightarrow \left[ \begin{array}{l}
\dfrac{1}{4} - x = \dfrac{9}{2}\\
\dfrac{1}{4} - x =  - \dfrac{9}{2}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{1}{4} - \dfrac{9}{2}\\
x = \dfrac{1}{4} - \left( { - \dfrac{9}{2}} \right)
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x =  - \dfrac{{17}}{4}\\
x = \dfrac{{19}}{4}
\end{array} \right.\\
*)\\
\left| {x + 1} \right| + \left| {x - 3} \right| = 0\\
\left| {x + 1} \right| \ge 0,\,\,\,\forall x\\
\left| {x - 3} \right| \ge 0,\,\,\,\forall x\\
 \Rightarrow \left| {x + 1} \right| + \left| {x - 3} \right| \ge 0,\,\,\forall x\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {x + 1} \right| = 0\\
\left| {x - 3} \right| = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x =  - 1\\
x = 3
\end{array} \right.\,\,\,\,\,\,\left( {vn} \right)\\
*)\\
\left| {x - 5} \right| + \left| {2 - y} \right| = 0\\
\left| {x - 5} \right| \ge 0,\,\,\forall x\\
\left| {2 - y} \right| \ge 0,\,\,\forall y\\
 \Rightarrow \left| {x - 5} \right| + \left| {2 - y} \right| \ge 0,\,\,\forall x,y\\
 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 5} \right| = 0\\
\left| {2 - y} \right| = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = 5\\
y = 2
\end{array} \right.\\
*)\\
{\left( {x + 1} \right)^{2018}} + {\left( {y + 2} \right)^{400}} = 0\\
{\left( {x + 1} \right)^{2018}} \ge 0,\,\,\forall x\\
{\left( {y + 2} \right)^{400}} \ge 0,\,\,\forall y\\
 \Rightarrow {\left( {x + 1} \right)^{2018}} + {\left( {y + 2} \right)^{400}} \ge 0,\,\,\,\forall x,y\\
 \Rightarrow \left\{ \begin{array}{l}
{\left( {x + 1} \right)^{2018}} = 0\\
{\left( {y + 2} \right)^{400}} = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x + 1 = 0\\
y + 2 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x =  - 1\\
y =  - 2
\end{array} \right.
\end{array}\)