, 2023

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 $(\frac{3}{2},0)$ is $3 + \frac{3}{2}$ which equals $\frac{6}{2} + \frac{3}{2}$ or $\frac{9}{2}$.