解:解方程$5m + 2x = \frac{1}{2} + x,$
移項(xiàng)得$2x - x = \frac{1}{2} - 5m,$
解得$x = \frac{1}{2} - 5m。$
解方程$x(m + 1) = m(x + 1),$
去括號(hào)得$mx + x = mx + m,$
移項(xiàng)得$mx + x - mx = m,$
解得$x = m。$
由題意得$\frac{1}{2} - 5m - m = 2,$
合并同類(lèi)項(xiàng)得$\frac{1}{2} - 6m = 2,$
移項(xiàng)得$-6m = 2 - \frac{1}{2},$
即$-6m = \frac{3}{2},$
解得$m = -\frac{1}{4}。$
答:當(dāng)$m = -\frac{1}{4}$時(shí),滿(mǎn)足條件。