Answer
5 mph during the first part, 6 mph for the second part
Work Step by Step
runs 5 miles, increases speed by 1 mph, then runs 2 additional miles, total time is 1 1/3 hours (4/3 hours)
$5/x + 2/(x+1) = 4/3$
$3x*(x+1)*(5/x + 2/(x+1) = 4/3)$
$3*(x+1)*5 + 2*3*x = 4(x)(x+1)$
$15(x+1) +6x=4(x^2+x)$
$15x+15 + 6x = 4x^2+4x$
$21x+15 = 4x^2+4x$
$21x+15-21x-15 = 4x^2+4x-21x-15$
$0=4x^2-17x-15$
$0=(4x+3)(x-5)$
$4x+3=0$
$4x+3-3=0-3$
$4x=-3$
$4x/4=-3/4$
$x=-3/4$ (not valid since we need time, which is a positive number)
$x-5=0$
$x=5$
$5+1$
$6$
