let's start by making the expression the recurring value, and then we'll multiply it by some power of 10 so that we move the recurring decimal over to the left, and then again by a power of 10 so we move a second part of the recurring decimal over, let's proceed,
[tex]\bf x = 0.\overline{5}~\hfill \begin{array}{llll} 10\cdot x&=&5.\overline{5}\\\\ 100\cdot x&=&55.\overline{5} \end{array}~\hfill \begin{array}{rllll} \stackrel{100x}{55.\overline{5}}\\\\ -\stackrel{10x}{5.\overline{5}}\\\cline{1-1} 50 \end{array} \\\\[-0.35em] ~\dotfill\\\\ 100x-10x=50\implies 90x=50\implies x=\cfrac{50}{90}\implies x=\cfrac{5}{9}\implies \stackrel{x = -0.\overline{5}}{x=-\cfrac{5}{9}}[/tex]
