In the diagram, the shaded region is bounded by the $x$-axis and the lines $y=x$, and $y=-2 x+3$. The area of the shaded region is
(A) $\frac{3}{4}$
(B) $\frac{3}{2}$
(C) $\frac{9}{4}$
(D) 1
(E) $\frac{\sqrt{10}}{4}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-093.jpg?height=364&width=467&top_left_y=493&top_left_x=1268)

Part B: Each correct answer is worth 6.
The bottom left vertex of the triangle has coordinates $(0,0)$, since $y=x$ (the line with positive slope) passes through the origin.

The bottom right vertex of the triangle corresponds with the $x$-intercept of the line $y=-2 x+3$, which we find by setting $y=0$ to obtain $-2 x+3=0$ or $x=\frac{3}{2}$. Thus, the bottom right vertex is $\left(\frac{3}{2}, 0\right)$.

The top vertex is the point of intersection of the two lines, which we find by combining the equations of the two lines to get $x=-2 x+3$ or $3 x=3$ or $x=1$.

Thus, this point of intersection is $(1,1)$.

Therefore, the triangle has a base along the $x$-axis of length $\frac{3}{2}$ and a height of length 1 (the $y$-coordinate of the top vertex).

Thus, the area of the triangle is $\frac{1}{2}\left(\frac{3}{2}\right)(1)=\frac{3}{4}$.