1. **State the problem:** Solve the equation $$\frac{2x+3}{x-1} = 4$$ for $x$. 2. **Formula and rules:** To solve a rational equation, multiply both sides by the denominator to eliminate the fraction, but remember to check for values that make the denominator zero (excluded values). 3. **Multiply both sides by the denominator:** $$\cancel{(x-1)} \cdot \frac{2x+3}{\cancel{x-1}} = 4 \cdot (x-1)$$ 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 $x$:** $$-2x = -7$$ $$x = \frac{-7}{-2} = \frac{7}{2}$$ 7. **Check for excluded values:** The denominator $x-1$ cannot be zero, so $x \neq 1$. Since $x=\frac{7}{2}$ is not excluded, it is a valid solution. **Final answer:** $$x = \frac{7}{2}$$