1. **Stating the problem:** Given the complex number $z=x-iy$, we want to find $x$ such that $$|z+3|-|z|=0.$$ 2. **Rewrite the problem:** This means $$|z+3| = |z|.$$ 3. **Write the complex numbers in terms of $x$ and $y$: ** $$z = x - iy,$$ so $$z + 3 = (x + 3) - iy.$$ 4. **Calculate magnitudes:** $$|z| = \sqrt{x^2 + y^2},$$ $$|z+3| = \sqrt{(x+3)^2 + y^2}.$$ 5. **Set the magnitudes equal:** $$\sqrt{(x+3)^2 + y^2} = \sqrt{x^2 + y^2}.$$ 6. **Square both sides to eliminate the square roots:** $$(x+3)^2 + y^2 = x^2 + y^2.$$ 7. **Cancel $y^2$ on both sides:** $$(x+3)^2 = x^2.$$ 8. **Expand the left side:** $$x^2 + 6x + 9 = x^2.$$ 9. **Subtract $x^2$ from both sides:** $$6x + 9 = 0.$$ 10. **Solve for $x$:** $$6x = -9,$$ $$x = -\frac{9}{6} = -\frac{3}{2}.$$ **Final answer:** $$x = -\frac{3}{2}.$$