Step-by-step Solution:

For an ellipse with its center at the origin, the equation of a directrix is given by \( x = -\frac{a}{e} \) (or \( +\frac{a}{e} \)).\nWe are given \( e = \frac{1}{2} \) and \( x = -4 \).\n\[ -\frac{a}{1/2} = -4 \implies 2a = 4 \implies a = 2 \]\n\nNow we find \( b^2 \) using the eccentricity formula \( b^2 = a^2(1 - e^2) \):\n\[ b^2 = 4 \left(1 - \left(\frac{1}{2}\right)^2\right) = 4 \left(1 - \frac{1}{4}\right) = 4 \times \frac{3}{4} = 3 \]\n\nThe equation of the ellipse is \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \).\n\nThe equation of the normal to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at a point \( (x_1, y_1) \) is:\n\[ \frac{a^2 x}{x_1} - \frac{b^2 y}{y_1} = a^2 - b^2 \]\n\nGiven the point \( (x_1, y_1) = (1, \frac{3}{2}) \):\n\[ \frac{4x}{1} - \frac{3y}{3/2} = 4 - 3 \]\n\[ 4x - 2y = 1 \]