How does 1/(1/4)x = 4/x?

1 / (1/4) x = 1 / (x / 4) .

The denominator (x / 4) gets inverted.

Hence 1 * ( 4 / x) = 4 / x

1
-----------
(1/4) x

4
----
x

Note: 1 / u = (u)^(-1), if "u" is any value.

[1 / (1 / 4)](1 / x)

= [(1 / 4)]^(-1) * (1 / x)

= 4 * (1 / x)

= 4 / x

In order for your statement to be true, you need some parentheses identifying the x as being part of the denominator.
1/((1/4)x) = 4*1/(4*(1/4)x) ... multiply numerator and denominator by 4
.. = 4/(1x)
.. = 4/x
_____
Divison is the name we give to multiplying by the multiplicative inverse. The multiplicative inverse is also called the reciprocal. Dividing by (1/4) is the same as multiplying by (4/1).
.. 1/(1/4) = 1*(4/1) = 4/1 = 4
If x starts in the denominator of your fraction, performing arithmetic on the numbers doesn't change its place in the denominator.
.. (1/(1/4))*(1/x) = 4*(1/x) = 4/x
_____
Another way to think about this is the denominator is 1/4 the size of the numerator, so will "go into it" 4 times. It is no different from 8/2 = 4 where the denominator is 1/4 of the numerator.

1/(1/4)x = 1/(x/4) = 4/x