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The problem in the image reads:

**Exercise 3**

Let \( f \) be a function that is strictly decreasing on \( \mathbb{R} \). If the curve \( C_f \) passes through the point \( A(6, -2) \), solve the inequality:
\[
f(x^2 - x) < -2.
\]

### Solution

1. Since \( f \) is a strictly decreasing function, \( f(x) < f(y) \) when \( x > y \).

2. The curve \( C_f \) passes through the point \( A(6, -2) \), which tells us that:
   \[
   f(6) = -2.
   \]

3. We need to solve the inequality:
   \[
   f(x^2 - x) < -2.
   \]

   Since \( f(6) = -2 \), we want \( x^2 - x > 6 \) to ensure \( f(x^2 - x) < f(6) = -2 \) due to the decreasing nature of \( f \).

4. Solve \( x^2 - x > 6 \):
   \[
   x^2 - x - 6 > 0.
   \]

5. Factorize the quadratic inequality:
   \[
   (x - 3)(x + 2) > 0.
   \]

6. Determine the intervals where this inequality holds:
   - The roots of \( (x - 3)(x + 2) = 0 \) are \( x = 3 \) and \( x = -2 \).
   - Use a sign chart or test intervals around these points:
     - For \( x < -2 \): \( (x - 3)(x + 2) > 0 \).
     - For \( -2 < x < 3 \): \( (x - 3)(x + 2) < 0 \).
     - For \( x > 3 \): \( (x - 3)(x + 2) > 0 \).

   Thus, \( (x - 3)(x + 2) > 0 \) when \( x < -2 \) or \( x > 3 \).

### Solution Set
The solution to the inequality \( f(x^2 - x) < -2 \) is:
\[
x \in (-\infty, -2) \cup (3, \infty).
\]

Would you like further details on any part of this solution, or have any questions?

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Here are some related questions for further understanding:

1. What happens if \( f \) is an increasing function instead?
2. How does knowing \( f(6) = -2 \) affect the solution?
3. How can we interpret \( f(x^2 - x) < -2 \) graphically?
4. What if the inequality were \( f(x^2 - x) > -2 \)?
5. How can we solve similar inequalities with other types of functions?

**Tip**: When dealing with inequalities involving monotonic functions, use their properties (increasing or decreasing) to determine the inequality's direction.

Let f be a strictly decreasing function on R. If the curve Cf passes through the point A(6, -2), solve the inequality f(x^2 - x) < -2.

Learn how to solve the inequality f(x^2 - x) < -2, where f is a strictly decreasing function. This problem involves factoring quadratic expressions and using properties of monotonic functions. Suitable for high school students in Grades 10-12.

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