What is the closest time between 7 and 8 when the hands of your watch are exactly
opposite each other?

At 7 the hour hand is  $$\frac{7}{12}\cdot360$$ = 210 degrees ahead of the minute hand.

Let after x minutes they are at 180 degrees with each other (since they are exact opposite)

Total angle covered by hour hand = 210 + 0.5x ( as hour hand moves at 0.5 degrees/minute)

Total angle covered by minute hand = 6x ( speed of minute hand is 6 degrees/minute)

As per the question

    Angle by hour hand - Angle by minute hand = 180

(210 + 0.5x) - 6x = 180

210 - 5.5x = 180

5.5x = 30

x  = 5.45 minutes

The closest answer is 5.5 minutes