To solve for $x$, let's analyze the given diagram.

Analysis:

1. The diagram shows a semicircle with a diameter labeled as $4 + x + 1 = x + 5$.
2. Inside the semicircle, there's a circle tangent to the diameter and the semicircle itself.
3. The distance $4$ and $1$ are the segments from the center of the semicircle to where the semicircle meets the diameter on either side of the circle.

Steps to solve:

1. Identify the radius of the semicircle:

   • The total diameter of the semicircle is $x + 5$, so the radius $R$ of the semicircle is $R = \frac{x + 5}{2}$.

2. Determine the radius $r$ of the inner circle:

   • The radius of the inner circle will be equal to $R$ minus the distance from the center of the semicircle to the point of tangency on the diameter. This distance is half of $4 + 1$, so it is $\frac{5}{2} = 2.5$.
   • Therefore, the radius $r$ of the inner circle is $R - r$.

3. Set up the equation:

   • Since the inner circle is tangent to the semicircle and the diameter, the distance from the center of the semicircle to the center of the inner circle is $R - r$.
   • The radius $r$ is also equal to $R - (R - 2.5)$.

   This results in the equation: $r = R - 2.5$

4. Substitute $R = \frac{x + 5}{2}$ into the equation: $r = \frac{x + 5}{2} - 2.5$

5. Solve for $r$ and match with the geometry: The radius $r$ is determined by the conditions of the problem, such as the circle fitting perfectly inside the semicircle with tangency.

   Therefore, we simplify and solve the equation: $\frac{x + 5}{2} - 2.5 = r$ The radius of the inner circle $r$ is calculated by the tangency condition.

Given the information, the equation simplifies to determine $x$.

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