Amountaineer climbed 1,000 feet at a rate of x feet per hour. he climbed an additional 5,000 feet at a different rate. this rate was 10 feet per hour less than twice the first rate. which expression represents the number of hours he climbed

Using distance=rate multiplied by time as the  formula, 

mountaineer climbed 1,000 feet at a rate of x feet per hour 
1000 = xt1 <--eq.1 

"He climbed an additional 5,000 feet at a different rate." 
5000 = x't2 <--eq.2 

"This rate was 10 feet per hour less than twice the first rate." 
x' = 2x - 10 <--eq.3 

We now have 3 equations/expressions to solve for the number of hours: 
From eq. 1 and 2: 
Total time = t1 +t2 = t 
t = 1000/x + 5000/x' 
but x' = 2x - 10 
t = 1000/x + 5000/(2x-10) 
t = 1000/x + 5000/[2(x-5)] 
t = 1000/x + 2500/(x-5) <--ANSWER