To solve for \( x \) in the triangle with angles measuring \( 75^\circ \), \( 41^\circ \), and \( (3x + 19)^\circ \), we begin by using the property that the sum of the angles in a triangle is always \( 180^\circ \).
Thus, we set up the equation:
\[ 75 + 41 + (3x + 19) = 180 \]
Now, let's simplify the left side of the equation:
\[ 75 + 41 = 116 \]
So the equation becomes:
\[ 116 + (3x + 19) = 180 \]
Combine the constant terms:
\[ 116 + 19 = 135 \]
This reduces our equation to:
\[ 135 + 3x = 180 \]
Next, subtract \( 135 \) from both sides:
\[ 3x = 180 - 135 \] \[ 3x = 45 \]
Now, divide both sides by 3 to solve for \( x \):
\[ x = \frac{45}{3} = 15 \]
Thus, the solution for \( x \) is:
\[ \boxed{15} \]