Đáp án:

$\begin{array}{l}
1)\frac{1}{4} + \frac{3}{4}x = \frac{5}{2}\\
 \Rightarrow \frac{{1 + 3x}}{4} = \frac{{10}}{4}\\
 \Rightarrow 1 + 3x = 10\\
 \Rightarrow 3x = 9\\
 \Rightarrow x = 3\\
2)\\
\left| {x - \frac{1}{4}} \right| = \frac{3}{8}\\
 \Rightarrow \left| {\frac{{8x}}{8} - \frac{2}{8}} \right| = \frac{3}{8}\\
 \Rightarrow \left| {\frac{{8x - 2}}{8}} \right| = \frac{3}{8}\\
 \Rightarrow \left| {8x - 2} \right| = 3\\
 \Rightarrow \left[ \begin{array}{l}
8x - 2 = 3\\
8x - 2 =  - 3
\end{array} \right. \Rightarrow \left[ \begin{array}{l}
8x = 5\\
8x =  - 1
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = \frac{5}{8}\\
x =  - \frac{1}{8}
\end{array} \right.\\
3)\\
\left( {x - \frac{3}{2}} \right):3\frac{1}{2} = \frac{4}{7}\\
 \Rightarrow \left( {x - \frac{3}{2}} \right):\frac{7}{2} = \frac{4}{7}\\
 \Rightarrow x - \frac{3}{2} = \frac{4}{7}.\frac{7}{2}\\
 \Rightarrow x - \frac{3}{2} = 2\\
 \Rightarrow x = 2 + \frac{3}{2}\\
 \Rightarrow x = \frac{7}{2}\\
4)\\
{8^x}:{4^x} = 4\\
 \Rightarrow {\left( {8:4} \right)^x} = 4\\
 \Rightarrow {2^x} = 4\\
 \Rightarrow {2^x} = {2^2}\\
 \Rightarrow x = 2\\
5)\\
\left| {x + \frac{2}{3}} \right| - 5 =  - 2\\
 \Rightarrow \left| {x + \frac{2}{3}} \right| = 3\\
 \Rightarrow \left[ \begin{array}{l}
x + \frac{2}{3} = 3\\
x + \frac{2}{3} =  - 3
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = \frac{7}{3}\\
x =  - \frac{{11}}{3}
\end{array} \right.\\
6)\\
\frac{{72 - x}}{{x + 40}} = \frac{7}{9}\left( {dkxd:x \ne  - 40} \right)\\
 \Rightarrow 9\left( {72 - x} \right) = 7\left( {x + 40} \right)\\
 \Rightarrow 9.72 - 9x = 7x + 7.40\\
 \Rightarrow 7x + 9x = 72.9 - 7.40\\
 \Rightarrow 16x = 368\\
 \Rightarrow x = 23
\end{array}$