To solve the equation \( |x + 2| = kx \) where \( k \in (0,1) \), we need to consider the definition of the absolute value function, which breaks the equation into two cases:

### Case 1: \( x + 2 \geq 0 \) (i.e., \( x \geq -2 \))
In this case, the absolute value function simplifies to:
\[
|x + 2| = x + 2
\]
So, the equation becomes:
\[
x + 2 = kx
\]
To solve for \( x \), subtract \( kx \) from both sides:
\[
x - kx = -2
\]
Factor out \( x \) on the left side:
\[
x(1 - k) = -2
\]
Now, solve for \( x \):
\[
x = \frac{-2}{1 - k}
\]

Since \( k \in (0, 1) \), \( 1 - k \) is positive, meaning \( \frac{-2}{1 - k} \) is negative. However, for this case, we assumed \( x \geq -2 \). Therefore, the solution for this case must satisfy:
\[
\frac{-2}{1 - k} \geq -2
\]
This inequality simplifies to:
\[
\frac{-2}{1 - k} \geq -2
\]
Multiply both sides by \( (1 - k) \) (note that \( 1 - k > 0 \)):
\[
-2 \geq -2(1 - k)
\]
Divide by \( -2 \) (this reverses the inequality):
\[
1 \leq 1 - k
\]
\[
k \leq 0
\]
But this contradicts the fact that \( k \in (0, 1) \). Thus, there is no valid solution in this case.

### Case 2: \( x + 2 < 0 \) (i.e., \( x < -2 \))
Here, the absolute value function simplifies to:
\[
|x + 2| = -(x + 2) = -x - 2
\]
So, the equation becomes:
\[
-x - 2 = kx
\]
Add \( x \) to both sides:
\[
-2 = kx + x
\]
Factor out \( x \) on the right side:
\[
-2 = x(k + 1)
\]
Solve for \( x \):
\[
x = \frac{-2}{k + 1}
\]

For this case, we assumed \( x < -2 \). Therefore, the solution must satisfy:
\[
\frac{-2}{k + 1} < -2
\]
This inequality simplifies to:
\[
\frac{-2}{k + 1} < -2
\]
Multiply both sides by \( (k + 1) \) (note that \( k + 1 > 0 \)):
\[
-2 < -2(k + 1)
\]
Divide by \( -2 \) (this reverses the inequality):
\[
1 > k + 1
\]
\[
k > 0
\]
This condition is satisfied since \( k \in (0, 1) \).

Thus, the only solution is:
\[
x = \frac{-2}{k + 1}
\]

### Final Answer:
The solution to the equation \( |x + 2| = kx \) where \( k \in (0,1) \) is:
\[
x = \frac{-2}{k + 1}
\]
under the condition \( k \in (0, 1) \).

Would you like more details on any part of the solution, or do you have any other questions?

---

1. How does the absolute value function affect the equation?
2. What happens if \( k \) is outside the interval \( (0, 1) \)?
3. Can you provide examples of solutions for specific values of \( k \)?
4. How does the inequality \( x < -2 \) influence the solution?
5. What if the equation had a different constant instead of 2?

**Tip:** Always check the constraints or conditions provided in a problem, as they can help simplify the solution or exclude invalid cases.

Learn how to solve the equation |x + 2| = kx where k is a constant in the interval (0,1). This detailed solution covers the application of absolute value, handling inequalities, and solving linear equations. Suitable for advanced high school students and beyond.

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