Biết: $\left| {\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}}} \right| - \dfrac{{\rm{1}}}{{\rm{2}}} = \dfrac{{\rm{1}}}{{\rm{4}}}$ khi đó $x$ nhận các giá trị là:

$\left| {\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}}} \right| - \dfrac{{\rm{1}}}{{\rm{2}}} = \dfrac{{\rm{1}}}{{\rm{4}}}$

$\left| {\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}}} \right| = \dfrac{{\rm{1}}}{{\rm{4}}} + \dfrac{{\rm{1}}}{{\rm{2}}}$

$\left| {\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}}} \right| = \dfrac{{\rm{1}}}{{\rm{4}}} + \dfrac{{\rm{2}}}{{\rm{4}}}$

$\left| {\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}}} \right| = \dfrac{{\rm{3}}}{{\rm{4}}}$

TH1: $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}} = \dfrac{{\rm{3}}}{{\rm{4}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{\rm{3}}}{{\rm{4}}} + \dfrac{{\rm{1}}}{{\rm{2}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{\rm{3}}}{{\rm{4}}} + \dfrac{{\rm{2}}}{{\rm{4}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{\rm{5}}}{{\rm{4}}}$

          ${\rm{x}} = \dfrac{{\rm{5}}}{{\rm{4}}}{\rm{:}}\dfrac{{\rm{3}}}{{\rm{4}}}$

          ${\rm{x}} = \dfrac{{\rm{5}}}{{\rm{4}}}{\rm{.}}\dfrac{{\rm{4}}}{{\rm{3}}}$

          ${\rm{x}} = \dfrac{{\rm{5}}}{{\rm{3}}}$

TH2: $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} - \dfrac{{\rm{1}}}{{\rm{2}}} = \dfrac{{ - {\rm{3}}}}{{\rm{4}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{ - {\rm{3}}}}{{\rm{4}}} + \dfrac{{\rm{1}}}{{\rm{2}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{ - {\rm{3}}}}{{\rm{4}}} + \dfrac{{\rm{2}}}{{\rm{4}}}$

         $\dfrac{{\rm{3}}}{{\rm{4}}}{\rm{x}} = \dfrac{{ - {\rm{1}}}}{{\rm{4}}}$

          ${\rm{x}} = \dfrac{{ - {\rm{1}}}}{{\rm{4}}}{\rm{:}}\dfrac{{\rm{3}}}{{\rm{4}}}$

          ${\rm{x}} = \dfrac{{ - {\rm{1}}}}{{\rm{4}}}{\rm{.}}\dfrac{{\rm{4}}}{{\rm{3}}}$

            ${\rm{x}} = \dfrac{{ - {\rm{1}}}}{{\rm{3}}}$