1. **Problem statement:** Solve the equation $$\sqrt{3} - x = x\sqrt{3} + x.$$\n\n2. **Rewrite the equation:** Move all terms to one side to isolate $x$:\n$$\sqrt{3} - x - x\sqrt{3} - x = 0.$$\n\n3. **Group like terms:** Combine the $x$ terms:\n$$\sqrt{3} - x(1 + \sqrt{3}) = 0.$$\n\n4. **Isolate $x$:**\n$$\sqrt{3} = x(1 + \sqrt{3}).$$\n\n5. **Solve for $x$:**\n$$x = \frac{\sqrt{3}}{1 + \sqrt{3}}.$$\n\n6. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate $1 - \sqrt{3}$:\n$$x = \frac{\sqrt{3}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{\sqrt{3} - 3}{1 - 3} = \frac{\sqrt{3} - 3}{-2} = \frac{3 - \sqrt{3}}{2}.$$\n\n**Final answer:** $$x = \frac{3 - \sqrt{3}}{2}.$$