To solve the equation \( |7 - 2x| = |5 - 3x| + |x + 2| \), we need to consider the different cases based on the values of \( x \) that affect the absolute value expressions.

### Step 1: Identify critical points

The critical points occur when the expressions inside the absolute values are equal to zero:

1. \( 7 - 2x = 0 \Rightarrow x = \frac{7}{2} = 3.5 \)
2. \( 5 - 3x = 0 \Rightarrow x = \frac{5}{3} \approx 1.67 \)
3. \( x + 2 = 0 \Rightarrow x = -2 \)

### Step 2: Set up intervals

The critical points divide the real line into the following intervals:

1. \( (-\infty, -2) \)
2. \( [-2, \frac{5}{3}) \)
3. \( [\frac{5}{3}, \frac{7}{2}) \)
4. \( [\frac{7}{2}, +\infty) \)

### Step 3: Analyze each interval

1. **Interval \( (-\infty, -2) \)**:
- Both \( |7 - 2x| = 7 - 2x \)
- \( |5 - 3x| = 5 - 3x \)
- \( |x + 2| = - (x + 2) = -x - 2 \)

The equation becomes:
\[
7 - 2x = (5 - 3x) + (-x - 2)
\]
Simplifying:
\[
7 - 2x = 5 - 3x - x - 2
\]
\[
7 - 2x = 3 - 4x
\]
\[
2x + 4x = 3 - 7
\]
\[
6x = -4 \Rightarrow x = -\frac{2}{3} \quad (\text{not in this interval})
\]

2. **Interval \( [-2, \frac{5}{3}) \)**:
- \( |7 - 2x| = 7 - 2x \)
- \( |5 - 3x| = 5 - 3x \)
- \( |x + 2| = x + 2 \)

The equation becomes:
\[
7 - 2x = (5 - 3x) + (x + 2)
\]
Simplifying:
\[
7 - 2x = 5 - 3x + x + 2
\]
\[
7 - 2x = 7 - 2x
\]
This holds true for all \( x \) in this interval.

3. **Interval \( [\frac{5}{3}, \frac{7}{2}) \)**:
- \( |7 - 2x| = 7 - 2x \)
- \( |5 - 3x| = 3x - 5 \)
- \( |x + 2| = x + 2 \)

The equation becomes:
\[
7 - 2x = (3x - 5) + (x + 2)
\]
Simplifying:
\[
7 - 2x = 3x - 5 + x + 2
\]
\[
7 - 2x = 4x - 3
\]
\[
7 + 3 = 4x + 2x
\]
\[
10 = 6x \Rightarrow x = \frac{5}{3} \quad (\text{in this interval})
\]

4. **Interval \( [\frac{7}{2}, +\infty) \)**:
- \( |7 - 2x| = 2x - 7 \)
- \( |5 - 3x| = 3x - 5 \)
- \( |x + 2| = x + 2 \)

The equation becomes:
\[
2x - 7 = (3x - 5) + (x + 2)
\]
Simplifying:
\[
2x - 7 = 3x - 5 + x + 2
\]
\[
2x - 7 = 4x - 3
\]
\[
-7 + 3 = 4x - 2x
\]
\[
-4 = 2x \Rightarrow x = -2 \quad (\text{not in this interval})
\]

### Conclusion

The solution is:

\[
\boxed{[-2, \frac{5}{3})}
\]

And the point \( x = \frac{5}{3} \) is also a valid solution. Therefore, the complete solution set is \( [-2, \frac{5}{3}] \).