Problem 6)
The reciprocal of any fraction is simply the flipped result of it. For example, the reciprocal of 2/3 is 3/2. The numerator and denominator swap.
In this case, 2/5 will turn into 5/2 after finding the reciprocal.
Answer: 5/2
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Problem 7)
Similar to problem 6, the reciprocal is -7/6 because the numerator and denominator swap places.
Answer: -7/6
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Problem 8)
Rewrite 7 as 7/1. This works because any number divided by 1 is itself. This is the rule x/1 = x for all real numbers x.
Now take the reciprocal of 7/1 to get 1/7
Answer: 1/7
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Problem 9)
Your answer is incorrect. We can check this by plugging in x = 3 to get...
(5/6)*x = 10
(5/6)*3 = 10
15/6 = 10
2.5 = 10
which is false
Here's how I would solve the problem
(5/6)*x = 10
6*(5/6)*x = 6*10
5x = 60
5x/5 = 60/5
x = 12
Alternatively, you can multiply both sides by the reciprocal of 5/6 to get the same result.
Check:
(5/6)*x = 10
(5/6)*12 = 10
60/6 = 10
10 = 10
Answer is confirmed
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Problem 10)
Multiply both sides by the reciprocal of -3/5 (which is -5/3) to get
6 = (-3/5)*x
6*(-5/3) = (-3/5)*x*(-5/3)
-30/3 = x
-10 = x
x = -10
Answer: x = -10
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Problem 11)
Multiply both sides by 3/2, which is the reciprocal of 2/3
(2/3)x = -3/5
(3/2)*(2/3)x = (3/2)*(-3/5)
x = (3*(-3))/(2*5)
x = -9/10
Answer: x = -9/10