What is the area of the triangle bounded by the lines y = 7 and y = 3 + 2|x - 4|?

To determine the area of the triangle formed by the lines ( y = 7 ) and ( y = 3 + 2|x - 4| ), we first need to find the points of intersection between these two lines.

The equation ( y = 3 + 2|x - 4| ) can be analyzed based on the expression inside the absolute value. There are two cases to consider:

1. For ( x \geq 4 ), the equation simplifies to ( y = 3 + 2(x - 4) = 2x - 5 ).

2. For ( x < 4 ), it simplifies to ( y = 3 + 2(4 - x) = 11 - 2x ).

Next, we find where each case intersects with ( y = 7 ).

For ( x \geq 4 ):

Set ( 2x - 5 = 7 ):

[

2x = 12 \implies x = 6

]

This gives the intersection point ( (6, 7) ).

For ( x < 4 ):

Set ( 11 - 2x =