math question (solving equation for x in terms of other letters)

Question

equation: 1/a - 1/x = 1/x - 1/b

Answer 1

1/a - 1/x = 1/x - 1/b

Add 1/b to both sides, add 1/x to both sides:

1/a + 1/b = 1/x + 1/x

Combine 1/x + 1/x

1/a + 1/b = 2/x

Take reciprocal of both sides

1/(1/a + 1/b) = x/2

Multiply both sides by 2:

2/(1/a + 1/b) = x

x = 2/(1/a + 1/b); a+b ≠ 0, ab ≠ 0

An equivalent form, found by multiplying the right hand side by (ab)/(ab) is

x = 2ab / (a+b); a+b ≠ 0, ab ≠ 0