Step 1 :Given that $\triangle MNP$ is a triangle, the sum of its interior angles is 180 degrees. Therefore, we can write the equation: $(4x - 3) + (9x - 6) + (6x - 1) = 180$
Step 2 :Combine like terms on the left side of the equation: $19x - 10 = 180$
Step 3 :Add 10 to both sides of the equation: $19x = 190$
Step 4 :Divide both sides of the equation by 19: $x = 10$
Step 5 :Substitute $x = 10$ into the expressions for $m \angle M, m \angle N,$ and $m \angle P$ to find the measures of these angles: $m \angle M = 4x - 3 = 4(10) - 3 = 37$ degrees, $m \angle N = 9x - 6 = 9(10) - 6 = 84$ degrees, $m \angle P = 6x - 1 = 6(10) - 1 = 59$ degrees
Step 6 :So, the value of $x$ is $\boxed{10}$, and the measures of the angles are $m \angle M = \boxed{37}$ degrees, $m \angle N = \boxed{84}$ degrees, and $m \angle P = \boxed{59}$ degrees