To solve for $x$ in this problem, we can analyze the information given:

1. The triangle $\triangle ABC$ has points $A$, $B$, $C$, and an additional point $D$ on $AC$ with segment lengths defined as follows:

   • $AB = x + 2$
   • $AD = x - 2$
   • $DC = x + 7$
   • $BC = 2x + 1$

2. Given that $\angle ADB$ and $\angle BDC$ are right angles, we can assume that the Pythagorean theorem will apply to triangles $\triangle ADB$ and $\triangle BDC$.

Let's proceed with these triangles separately.

Step 1: Apply the Pythagorean Theorem on $\triangle ADB$

For $\triangle ADB$: $AB^2 = AD^2 + BD^2$ Substitute the given values: $(x + 2)^2 = (x - 2)^2 + BD^2$ Expanding and solving for $BD$: $(x + 2)^2 = x^2 + 4x + 4$ \