The problem reads as follows:

"△STU is an equilateral triangle. If ST is one less than twice $x$, SU is 37 less than five times $x$, and TU is 11 more than $x$, find $x$ and the measure of each side."

Step-by-step solution:

1. Equilateral Triangle Properties:
   Since △STU is equilateral, all sides are equal. Therefore: $ST = SU = TU$

2. Equations for the sides:
   According to the problem: $ST = 2x - 1, \quad SU = 5x - 37, \quad TU = x + 11$

   Since all the sides are equal, we can set up the following system of equations: $2x - 1 = 5x - 37 \quad \text{(1)}$ $5x - 37 = x + 11 \quad \text{(2)}$

3. Solve the system of equations:

   • Start with equation (1): $2x - 1 = 5x - 37$ Simplifying this: $-1 + 37 = 5x - 2x \quad \Rightarrow \quad 36 = 3x \quad \Rightarrow \quad x = 12$

   • Now substitute $x = 12$ into equation (2): $5(12) - 37 = 12 + 11$ Simplifying: $60 - 37 = 23 \quad \Rightarrow \quad 23 = 23$ This confirms that $x = 12$ is correct.

4. Find the length of each side:
   Now that we know $x = 12$, substitute it back into the expressions for the sides: $ST = 2(12) - 1 = 24 - 1 = 23$ $SU = 5(12) - 37 = 60 - 37 = 23$ $TU = 12 + 11 = 23$

Thus, the value of $x$ is 12, and the measure of each side of the equilateral triangle is 23 units.

Would you like more details or have any further questions?

Here are 5 related questions for practice:

1. What are the properties of an equilateral triangle?
2. How would you solve for $x$ if the triangle was not equilateral?
3. Can you find the perimeter of the triangle with the given side lengths?
4. How would you adjust the problem if one side was described as three less than twice $x$?
5. Can you derive a general formula for the sides if the triangle is equilateral but described differently?

Tip: Always check for symmetry in geometry problems like this one when dealing with equilateral or isosceles triangles!