Triangles S U V and R U W are connected at point U. Angles S U V and W U T are right angles. The length of hypotenuse S V is 2 x + 9 and the length of hypotenuse W T is 4 x minus 1. Sides V U and U W are congruent.

To solve for the value of \(x\) that would make triangle \(S U V\) congruent to triangle \(T U M\) by the Hypotenuse-Leg (HL) theorem, we first need to analyze the information given.

1. Understanding the triangles:

   • We know that triangle \(S U V\) has a hypotenuse \(S V = 2x + 9\).
   • Triangle \(R U W\) has a hypotenuse \(W T = 4x - 1\).
   • The legs \(V U\) and \(U W\) are congruent, meaning that \(V U = U W\).

2. Using the HL theorem:

   • The HL theorem states that in a right triangle, if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of another triangle, then the two triangles are congruent.
   • In this case, we have \(SV\) and \(WT\) as the hypotenuses and \(VU\) and \(UW\) as the legs.

3. Setting up the equations:

   • From the congruence of the hypotenuses, we set: \[ 2x + 9 = 4x - 1 \]
   • Solving for \(x\): \[ 2x + 9 = 4x - 1 \implies 9 + 1 = 4x - 2x \implies 10 = 2x \implies x = 5 \]

4. Verification of legs:

   • Since \(VU\) and \(UW\) are congruent, we need to confirm that at \(x = 5\), these lengths still hold for triangle congruence:
   • There is no specific length given for \(VU\) and \(UW\), but since we do not calculate them because we are looking directly for the value of \(x\) that maintains the congruence based on the hypotenuses, we will assume that verification can be assumed since the problem statement does not indicate otherwise.

Thus, the answer is: \[ \boxed{5} \]