At 2: 00p. m., joon and gerace start driving toward each other from points 174 miles apart. they meet at 4: 00 p. m. joon's speed was 7 miles per hour greater than grace's speed. find their speeds.

Their speeds are  ' s '  and  ' s+7 ' .

Since Joon is decreasing the size of the distance between them from one end,
and Gerace is decreasing it from the other end, the distance between them is
shrinking, and they meet when the SUM of their distances is 174 miles.

The sum of their speeds is (s) + (s+7) = (2s+7) m/h

They meet after driving for 2 hours, so 

                               (2h) x (2s+7) m/h = 174 miles

Eliminate the parentheses:          (4s + 14) = 174

Subtract  14  from each side:       4s  =  160

Divide each side by  4 :                 s  =  40

Their speeds are  40 mph  and  47 mph .

Check:

40 mph for 2 hours = 80 miles
47 mph for 2 hours = 94 miles  

80 + 94 = 174 miles        yay !