Step 1 :Rearrange the equation to group terms with $x$ and $y$ on one side and constants on the other side, giving $3y + 4x - 3x = 2 + 2$
Step 2 :Simplify the equation to $y + x = 4$
Step 3 :Solve for $y$ by subtracting $x$ from both sides, giving $y = 4 - x$
Step 4 :To find the $x$-intercept, set $y = 0$ and solve for $x$, giving $x = \boxed{4}$