Final answer:
The area of the triangular region in question is found by identifying the points of intersection of the lines, determining the base and height of the triangle, and applying the area formula for a triangle. The correct answer is 12 square units.
Step-by-step explanation:
The area of the triangular region enclosed by the x-axis and the lines y = x and y = -2x + 12 can be found by determining the points of intersection of the lines and then using the formula for the area of a triangle. To find the points of intersection, we set y = x equal to y = -2x + 12, leading to 3x = 12 and x = 4. Similarly, setting y = 0 in both equations, we find that the x-intercepts are (0,0) for y = x and (6,0) for y = -2x + 12.
With the vertices of the triangle at (0,0), (4,4), and (6,0), we can use the base and height to find the area. The base of the triangle is the distance between the x-intercepts, which is 6 units, and the height is the y-coordinate of the vertex (4,4), which is 4 units. The area is then calculated as 1/2 × base × height, yielding 1/2 × 6 × 4 = 12 square units.
Therefore, the correct answer is c. 12 square units.