The number of real solutions of |$x-3|=2x+1$ is:

If $x$ ≥ 3 the equation reads $x-3=2x+1$, giving $x=-4$, which contradicts $x$ ≥ 3. If $x$ < 3 it reads $3-x=2x+1$, giving $3x=2$ and $x=2/3$, which is valid. So there is exactly one solution: |2/3 − 3| = 7/3 and $2(2/3)+1=7/3$.