Find the slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \). Then find an equation for the line tangent The slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \) is (Type an integer or a simplified fraction.)

We start with the function \[ g(x)=\frac{3x}{x-2}. \] **Step 1. Differentiate \( g(x) \) using the quotient rule** The quotient rule states that if \[ h(x)=\frac{f(x)}{u(x)}, \] then \[ h'(x)=\frac{u(x)f'(x)-f(x)u'(x)}{[u(x)]^2}. \] For \( g(x)=\frac{3x}{x-2} \), we have: \[ f(x)=3x,\quad f'(x)=3, \] \[ u(x)=x-2,\quad u'(x)=1. \] Thus, \[ g'(x)=\frac{(x-2)(3)-3x(1)}{(x-2)^2}. \] **Step 2. Simplify the derivative** Multiply out the numerator: \[ (x-2)(3)=3x-6, \] so, \[ g'(x)=\frac{3x-6-3x}{(x-2)^2}=\frac{-6}{(x-2)^2}. \] **Step 3. Evaluate the derivative at \( x=4 \)** Substitute \( x=4 \) into the derivative: \[ g'(4)=\frac{-6}{(4-2)^2}=\frac{-6}{2^2}=\frac{-6}{4}=-\frac{3}{2}. \] Thus, the slope of the graph at \( (4,6) \) is \[ -\frac{3}{2}. \] **Step 4. Find the equation of the tangent line** The point-slope form of the line is given by \[ y-y_{1}=m(x-x_{1}), \] where \( m=-\frac{3}{2} \) and \( (x_{1},y_{1})=(4,6) \). Substitute the values: \[ y-6=-\frac{3}{2}(x-4). \] This is the equation of the tangent line. To express it in slope-intercept form, solve for \( y \): \[ y = -\frac{3}{2}(x-4) + 6, \] \[ y = -\frac{3}{2}x + 6 + 6, \] \[ y = -\frac{3}{2}x + 12. \] The slope of the graph at \( (4,6) \) is \[ -\frac{3}{2}. \]