Phương trình $\dfrac{1}{3} - \left| {\dfrac{5}{4} - 2x} \right| = \dfrac{1}{4}$ có nghiệm là

$\begin{array}{l}\;\;\;\;\;\dfrac{1}{3} - \left| {\dfrac{5}{4} - 2x} \right| = \dfrac{1}{4}\\ \Leftrightarrow \left| {\dfrac{5}{4} - 2x} \right| = \dfrac{1}{3} - \dfrac{1}{4}\\ \Leftrightarrow \left| {\dfrac{5}{4} - 2x} \right| = \dfrac{1}{{12}}\,\left( * \right)\end{array}$

$\begin{array}{l}TH1:\;\dfrac{5}{4} - 2x \ge 0 \Leftrightarrow x \le \dfrac{5}{8} \Rightarrow \left| {\dfrac{5}{4} - 2x} \right| = \dfrac{5}{4} - 2x\\ \Rightarrow pt\,\,\left( * \right) \Leftrightarrow \dfrac{5}{4} - 2x = \dfrac{1}{{12}}\\ \Leftrightarrow 2x = \dfrac{7}{6}\\ \Leftrightarrow x = \dfrac{7}{{12}}\;\;\left( {tm} \right)\end{array}$              

$\begin{array}{l}TH2:\;\;\dfrac{5}{4} - 2x < 0 \Leftrightarrow x > \dfrac{5}{8} \Rightarrow \left| {\dfrac{5}{4} - 2x} \right| =  - \dfrac{5}{4} + 2x\;\\ \Rightarrow pt\,\left( * \right) \Leftrightarrow  - \dfrac{5}{4} + 2x = \dfrac{1}{{12}}\\ \Leftrightarrow 2x = \dfrac{4}{3}\\ \Leftrightarrow x = \dfrac{2}{3}\;\;\left( {tm} \right).\;\end{array}$

Vậy phương trình có hai nghiệm $x = \dfrac{7}{{12}}$ và $x = \dfrac{2}{3}$.