ANSWER
She averaged at 32 mph for 5 hours and 39 mph for 2 hours.
EXPLANATION
Let
![x](https://tex.z-dn.net/?f=x)
be the number of hours she travelled before noon and
![y](https://tex.z-dn.net/?f=y)
be the number of hours she travelled after noon.
Then since she travelled for a total of 7 hours, we can write the equation,
![x + y = 7.....eqn(1)](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%207.....eqn%281%29)
We were also given that, before noon,she averaged at 32 mph.
We know that,
![speed = \frac{distance}{time \: taken}](https://tex.z-dn.net/?f=speed%20%3D%20%20%5Cfrac%7Bdistance%7D%7Btime%20%5C%3A%20taken%7D%20)
Let
![d_1](https://tex.z-dn.net/?f=d_1)
be the distance before noon.
![\Rightarrow \: 32 = \frac{d_1}{x}](https://tex.z-dn.net/?f=%5CRightarrow%20%5C%3A%2032%20%3D%20%5Cfrac%7Bd_1%7D%7Bx%7D%20)
This implies that,
![d_1 = 32x](https://tex.z-dn.net/?f=d_1%20%3D%2032x)
Also let the distance she covered after noon be,
![d_2](https://tex.z-dn.net/?f=d_2)
Then,
![39 = \frac{d_2}{y}](https://tex.z-dn.net/?f=%2039%20%3D%20%20%5Cfrac%7Bd_2%7D%7By%7D%20)
This implies that,
![d_2 = 39y](https://tex.z-dn.net/?f=d_2%20%3D%2039y)
Since she covered a total distance of 238 miles, we can write the equation,
![d_1 + d_2 = 238](https://tex.z-dn.net/?f=d_1%20%2B%20d_2%20%3D%20238)
This means that,
![32x + 39y = 238....eqn(2)](https://tex.z-dn.net/?f=32x%20%2B%2039y%20%3D%20238....eqn%282%29)
We multiply equation by 32 to get,
![32x + 32y = 224...eqn(3)](https://tex.z-dn.net/?f=32x%20%2B%2032y%20%3D%20224...eqn%283%29)
Equation (2) minus equation (3), will give us,
![7y = 14](https://tex.z-dn.net/?f=7y%20%3D%2014)
This implies that,
![y = 2](https://tex.z-dn.net/?f=y%20%3D%202)
We substitute the value of y into equation (1) to get,
![x + 2 = 7](https://tex.z-dn.net/?f=x%20%2B%202%20%3D%207)
![x = 7 - 2](https://tex.z-dn.net/?f=x%20%3D%207%20-%202)
![x = 5](https://tex.z-dn.net/?f=x%20%3D%205)
We can now conclude that she averaged at 32 mph for 5 hours and 39 mph for 2 hours.