Geometry: Common Core (15th Edition)

$x = 20$
If three parallel lines cut through two transversals, then the segments on one transversal that was cut by the parallel lines are proportional to the segments on the other transversal that was cut by the parallel lines. Now we can set the proportions to compare the segments on one transversal to the segments on the other transversal: $\frac{4x}{5x} = \frac{4x + 8}{6x - 10}$ Use the cross product property to get rid of the fractions: $4x(6x - 10) = 5x(4x + 8)$ Use distribution first: $24x^2 - 40x = 20x^2 + 40x$ Move all terms to the left side of the equation: $24x^2 - 20x^2 - 40x - 40x = 0$ Combine like terms: $4x^2 - 80x = 0$ Factor out $4$ from each side of the equation: $x^2 - 20x = 0$ Factor out $x$ from the left side of the equation: $x(x - 20) = 0$ Set each factor equal to $0$: $x = 0$ or $x - 20 = 0$ We have to discard $x = 0$ because we cannot have a length of $0$. For the other factor, add $20$ to each side of the equation to solve for $x$: $x = 20$