$${{\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet}\atop O \;\quad\quad M\quad\quad\; P}$$
Given that $OM = x + 8$, $MP = 2x - 6$, $OP = 44$, is $M$ the midpoint of $OP$?
$${{\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet}\atop O \;\quad\quad M\quad\quad\; P}$$
Given that $OM = x + 8$, $MP = 2x - 6$, $OP = 44$, is $M$ the midpoint of $OP$?
HINT $$OM + MP = OP$$ Move your mouse over the gray area for the complete solution.
Note that $$OM + MP = OP$$ and hence we get that $3x+2 = 44 \implies x =14$. Hence, $OM = 14 + 8 = 22 = \dfrac{44}2 = \dfrac{OP}2$. Hence, $M$ is indeed the midpoint.
We have that $$OM + MP = OP$$ so $$(x+8)+(2x-6)=3x+2=44$$ $$3x=42$$ $$x=14$$ Therefore $OM=x+8=22$ and $MP=2\cdot 14-6=28-6=22$. Because the lengths of $OM$ and $MP$ are equal, $M$ is the midpoint of the line $OP$.