1. **Problem statement:** We have two affine functions $h(x) = -\frac{5}{2}x + 6$ and $g(x)$ passing through points $B(6,0)$ and $C(0,3)$. Points $A(\frac{3}{2},0)$ and $D(0,6)$ are intersections of $h$ with axes. Point $E$ lies on both graphs with abscissa $\frac{3}{2}$. We want to find the area of the shaded polygon bounded by points $D, C, E, A, B$. 2. **Find $g(x)$ equation:** Using points $B(6,0)$ and $C(0,3)$: Slope $m_g = \frac{0-3}{6-0} = -\frac{3}{6} = -\frac{1}{2}$. Equation form: $g(x) = mx + b$. Using point $C(0,3)$: $b = 3$. So, $g(x) = -\frac{1}{2}x + 3$. 3. **Find point $E$ coordinates:** Given $x_E = \frac{3}{2}$, find $y_E$ by substituting into $g(x)$ or $h(x)$ (should be equal): $$ y_E = g\left(\frac{3}{2}\right) = -\frac{1}{2} \times \frac{3}{2} + 3 = -\frac{3}{4} + 3 = \frac{9}{4} = 2.25 $$ Check with $h(x)$: $$ h\left(\frac{3}{2}\right) = -\frac{5}{2} \times \frac{3}{2} + 6 = -\frac{15}{4} + 6 = -3.75 + 6 = 2.25 $$ So, $E\left(\frac{3}{2}, \frac{9}{4}\right)$. 4. **List polygon vertices in order:** $D(0,6), C(0,3), E\left(\frac{3}{2}, \frac{9}{4}\right), A\left(\frac{3}{2}, 0\right), B(6,0)$. 5. **Calculate area using Shoelace formula:** Shoelace formula for polygon with vertices $(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$: $$ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| $$ where $(x_{n+1}, y_{n+1}) = (x_1, y_1)$. Calculate terms: $$ \begin{aligned} \sum x_i y_{i+1} &= 0 \times 3 + 0 \times \frac{9}{4} + \frac{3}{2} \times 0 + \frac{3}{2} \times 0 + 6 \times 6 \\ &= 0 + 0 + 0 + 0 + 36 = 36 \\ \sum y_i x_{i+1} &= 6 \times 0 + 3 \times \frac{3}{2} + \frac{9}{4} \times \frac{3}{2} + 0 \times 6 + 0 \times 0 \\ &= 0 + \frac{9}{2} + \frac{27}{8} + 0 + 0 = \frac{36}{8} + \frac{27}{8} = \frac{63}{8} = 7.875 \end{aligned} $$ 6. **Compute area:** $$ \text{Area} = \frac{1}{2} |36 - 7.875| = \frac{1}{2} \times 28.125 = 14.0625 $$ 7. **Final answer:** The area of the shaded region is $14.0625$ square units.