1. **Stating the problem:** We are given the equation $$\frac{2x+3}{x-1} = 4$$ and need to solve for $x$. 2. **Formula and rules:** To solve rational equations 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 $x-1$:** $$\cancel{\frac{2x+3}{x-1}} \times (x-1) = 4 \times \cancel{(x-1)}$$ which simplifies to $$2x + 3 = 4(x - 1)$$ 4. **Expand the right side:** $$2x + 3 = 4x - 4$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$2x - 4x = -4 - 3$$ 6. **Simplify both sides:** $$-2x = -7$$ 7. **Divide both sides by $-2$ to solve for $x$:** $$\frac{-2x}{\cancel{-2}} = \frac{-7}{\cancel{-2}}$$ which gives $$x = \frac{7}{2}$$ 8. **Check the solution:** Since $x = \frac{7}{2} \neq 1$, it does not make the denominator zero, so it is valid. **Final answer:** $$x = \frac{7}{2}$$