Did I solve this right? Hope you can analyze and tell me where I went wrong or how you would solve it.

(1/a)-(1/x) = (1/x)-(1/b)
xb-ab = ab-xa
xb+xa = ab+ab
(xb+xa)/x = (2ab)/x
x(b+a) = (2ab)/x
x = (2ab)/(b+a)

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

did it a little differently but you got the right answer.

The third line of solution is correct.
Fourth line is also correct, but why would you divide both sides by x?
Fifth line is where everything goes wrong. You've completely gotten rid of x in denominator, that you put there in previous line.

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Just because you're factoring out x in numerator doesn't mean you can just get rid of x
Sixth line, everything goes wrong again. This time you've gotten rid of x in denominator on right side of equation. But the funny thing is that in this case two wrongs do make a right.

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Since you made the same mistake once on left side of equation, then for next step on right side of equation, you end up with correct value for x.

This solution would not get you full marks on a test.

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My solution:

(1/a)-(1/x) = (1/x)-(1/b)
xb-ab = ab-xa
xb+xa = ab+ab
----------------------------… eliminate 4th line altogether
x(b+a) = (2ab) --------> factor out x on left side. NO x on right side
x = (2ab)/(b+a)

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