Calculate Negative Area: Finding Area Under y=-(x-4)² + 9

To solve this problem, we'll follow these steps:

• Step 1: Set up the inequality $-(x-4)^2 + 9 < 0$.
• Step 2: Simplify to get $(x-4)^2 > 9$.
• Step 3: Solve $(x-4)^2 > 9$ into two linear inequalities.
• Step 4: Identify the solution intervals.

Let's work through these steps:

Step 1: Set up the inequality:
We have $-(x-4)^2 + 9 < 0$.

Step 2: Simplify the inequality:
This becomes $(x-4)^2 > 9$.

Step 3: Solve the quadratic inequality:
The inequality $(x-4)^2 > 9$ can be split into two cases:
$x-4 > 3$ or $x-4 < -3$.

Simplifying these gives:
$x > 7$ or $x < 1$.

Step 4: Identify the solution intervals:
Thus, the intervals where the function is negative are $x < 1$ or $x > 7$.

Therefore, the correct solution is the interval $x < 1 , 7 < x$.

This matches choice 3.