Chapter 4 - Proportions, Percents, and Solving Inequalities - 4.1 - Ratios and Proportions - Problem Set 4.1 - Page 149: 30

$x=-10$

Using cross-multiplication and the properties of equality, the value of the variable that satisfies the given equation, $ \dfrac{x-2}{6}=\dfrac{x-6}{8} ,$ is \begin{array}{l}\require{cancel} 8(x-2)=6(x-6) \\\\ 8x-16=6x-36 \\\\ 8x-6x=-36+16 \\\\ 2x=-20 \\\\ x=-\dfrac{20}{2} \\\\ x=-10 .\end{array}