1. **Stating the problem:** Solve the equation $$\frac{2x+3}{x-1} = 4$$ for $x$. 2. **Formula and rules:** To solve a rational equation like this, multiply both sides by the denominator to eliminate the fraction, but remember $x \neq 1$ because the denominator cannot be zero. 3. **Multiply both sides by $x-1$:** $$\cancel{\frac{2x+3}{x-1}} \times (x-1) = 4 \times (x-1)$$ which simplifies to $$2x + 3 = 4(x - 1)$$ 4. **Expand the right side:** $$2x + 3 = 4x - 4$$ 5. **Bring all terms to one side:** $$2x + 3 - 4x + 4 = 0$$ which simplifies to $$-2x + 7 = 0$$ 6. **Solve for $x$:** $$-2x = -7$$ $$x = \frac{-7}{-2}$$ $$x = \frac{7}{2}$$ 7. **Check for restrictions:** Since $x \neq 1$, and $\frac{7}{2} = 3.5 \neq 1$, the solution is valid. **Final answer:** $$x = \frac{7}{2}$$