https://file.aiquickdraw.com/mathbot/file/172728204304936vqkyt2.jpg

The equation given in the image is:

\[
e^{3x} - 3x + 3 = 0
\]

We are tasked with determining how many solutions this equation admits in the set of real numbers \(\mathbb{R}\).

### Approach:

1. **Behavior of \( e^{3x} \)**:
   - The function \( e^{3x} \) is an exponential function, which grows very rapidly as \( x \to \infty \) and tends to 0 as \( x \to -\infty \). It is always positive for all real values of \( x \).

2. **Behavior of the linear term \( -3x + 3 \)**:
   - The function \( -3x + 3 \) is a linear function, decreasing as \( x \) increases.

3. **Finding intersections**:
   - Since the two terms represent functions with very different growth behaviors (one exponential and one linear), we expect that the equation may have one or more intersections (solutions).
   
4. **Graphical Interpretation**:
   - The rapid growth of \( e^{3x} \) and the linear nature of \( -3x + 3 \) suggest that there will be at least one point where the exponential function overtakes the linear function.
   - Numerical or graphical solutions indicate that there are **two points** where the functions intersect.

Thus, the number of solutions is **2**.

### Final Answer: C) 2

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Would you like further details or have any questions? Here are some related problems:

1. Solve \( e^{2x} - 4x = 0 \) for \( x \in \mathbb{R} \).
2. How would the solution change if the equation was \( e^{2x} - x + 1 = 0 \)?
3. Discuss the behavior of \( e^{kx} - x + c = 0 \) as \( k \) and \( c \) vary.
4. What happens if we replace \( e^{3x} \) with \( e^{-3x} \) in the equation?
5. How many solutions does \( e^{x} - x^2 = 0 \) admit in \( \mathbb{R} \)?

**Tip:** When combining exponential and polynomial terms in an equation, the number of solutions can often be predicted by studying the growth rates of the individual terms.

How many solutions, in ℝ, does the equation e^{3x} - 3X + 3 = 0 admit?

Learn how to determine the number of real solutions for the equation e^3x - 3x + 3 = 0. This problem covers key concepts like exponential functions and linear equations, suitable for students in Grades 10-12.

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