Answer
5.73 mph for 5 miles, 6.73 mph for 2 miles
Work Step by Step
5 miles at a rate of $x$ mph
2 miles at a rate of $x+1$ mph
$(5/x) + (2/(x+1)) = 4/3$
$\frac{5}{x} + \frac{2}{x+1} = \frac{4}{3}$
$x*(x+1)*3*\frac{5}{x} + x*(x+1)*3*\frac{2}{x+1} = x*(x+1)*3*\frac{4}{3}$
$\frac{x*(x+1)*3*5}{x} + \frac{x*(x+1)*3*2}{x+1} = \frac{x*(x+1)*3*4}{3}$
$\frac{(x+1)*3*5}{1} + \frac{x*3*2}{1} = \frac{x*(x+1)*4}{1}$
$15(x+1) + 6x = 4x(x+1)$
$15x+15+6x=4x^2+4$
$21x + 15=4x^2+4$
$4x^2+4=21x+15$
$4x^2-21x-11=0$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(-21)±\sqrt {(-21)^2-4*4*(-11)})/2*4$ $x=(21±\sqrt {441+4*4*(11)})/8$
$x=(21±\sqrt {441+176)})/8$
$x=(21±\sqrt {617})/8$
We want the positive square root since we are talking about time. $x=(21±\sqrt {617})/8$
$x=(21±24.8395)/8$
$x=(21+24.8395)/8$
$x=(45.8395)/8$
$x=5.73$