1. The problem is to solve the equation $$\frac{2x+3}{x-1} = 4.$$\n\n2. We use the property of equations that if $$\frac{A}{B} = C,$$ then $$A = B \times C,$$ provided $$B \neq 0.$$\n\n3. Multiply both sides of the equation by $$x-1$$ to eliminate the denominator:\n$$2x + 3 = 4(x - 1).$$\n\n4. Expand the right side:\n$$2x + 3 = 4x - 4.$$\n\n5. Bring all terms involving $$x$$ to one side and constants to the other:\n$$2x - 4x = -4 - 3,$$\nwhich simplifies to\n$$-2x = -7.$$\n\n6. Divide both sides by $$-2$$ to solve for $$x$$:\n$$x = \frac{-7}{-2} = \frac{7}{2} = 3.5.$$\n\n7. Check the solution by substituting $$x = 3.5$$ back into the original equation to ensure the denominator is not zero and the equality holds.\n\nFinal answer: $$x = \frac{7}{2}.$$