The line with a slope of 2 and $y$-intercept 6 has equation $y=2x+6$. To find its $x$-intercept, we set $y=0$ to obtain $0=2x+6$ or $2x=-6$, which gives $x=-3$. The line with a slope of -4 and $y$-intercept 6 has equation $y=-4x+6$. To find its $x$-intercept, we set $y=0$ to obtain $0=-4x+6$ or $4x=6$, which gives $x=\frac{6}{4}=\frac{3}{2}$. The distance between the points on the $x$-axis with coordinates $(-3,0)$ and $\left(\frac{3}{2}, 0\right)$ is $3+\frac{3}{2}$ which equals $\frac{6}{2}+\frac{3}{2}$ or $\frac{9}{2}$.