Solution:
(i) 1/4! + 1/5! = x/6!
We can write 4! and 5! as:
5! = 5 × 4 × 3 × 2 × 1
6! = 6 × 5 × 4 × 3 × 2 × 1
So by making use of these values,
$ 1/4!\text{ }+\text{ }1/5!\text{ }=\text{ }x/6! $
$ 1/4!\text{ }+\text{ }1/\left( 5\times 4! \right)\text{ }=\text{ }x/6! $
$ \left( 5\text{ }+\text{ }1 \right)\text{ }/\text{ }\left( 5\times 4! \right)\text{ }=\text{ }x/6! $
$ 6/5!\text{ }=\text{ }x/\left( 6\times 5! \right) $
$ x\text{ }=\text{ }\left( 6\text{ }\times \text{ }6\text{ }\times \text{ }5! \right)/5! $
$ x=\text{ }36 $
Therefore, the value of x is 36.
(ii) x/10! = 1/8! + 1/9!
We know that we can write:
10! = 10 × 9!
9! = 9 × 8!
So by making use of these values,
$ x/10!\text{ }=\text{ }1/8!\text{ }+\text{ }1/9! $
$ x/10!\text{ }=\text{ }1/8!\text{ }+\text{ }1/\left( 9\times 8! \right) $
$ x/10!\text{ }=\text{ }\left( 9\text{ }+\text{ }1 \right)\text{ }/\text{ }\left( 9\times 8! \right) $
$ x/10!\text{ }=\text{ }10/9! $
$ x/\left( 10\times 9! \right)\text{ }=\text{ }10/9! $
$ x\text{ }=\text{ }\left( 10\text{ }\times \text{ }10\text{ }\times \text{ }9! \right)/9! $
$ x=\text{ }10\text{ }\times \text{ }10 $
$ x=\text{ }100 $
Therefore, the value of x is 100.