Chapter 1 - Section 1.1 - Linear Equations - 1.1 Assess Your Understanding - Page 90: 51

$x=2$

$\displaystyle \begin{array}{{>{\displaystyle}l}} We\ have:\ \frac{x}{x-2} +3=\frac{2}{x-2}\\ \\ First,\ multiply\ each\ side\ by\ the\ square\ of\ the\ denominator:\\ \left(\frac{x}{x-2} +3\right)( x-2)^{2} =\frac{2}{x-2}( x-2)^{2}\\ \\ Now,\ expand\ and\ simplify:\\ \\ \frac{x( x-2)^{2}}{x-2} +3( x-2)^{2} =2( x-2)\\ \\ x( x-2) +3( x-2)^{2} =2x-4\\ \\ x^{2} -2x+3\left( x^{2} -4x+4\right) =2x-4\\ \\ x^{2} -2x+3x^{2} -12x+12=2x-4\\ \\ 4x^{2} -16x+16=0\\ \\ x^{2} -4x+4=0\\ \\ ( x-2)^{2} =0\\ \\ x=2\\ \\ \\ \\ \end{array}$