Question

solve foe x : 1/abx = 1/a+ 1/b + 1/x

Answer 1

$1 \div abx = 1 \div a + 1 \div b + 1 \div x \\ 1 \div abx = (bx + ax + ab) \div abx \\ 1 = bx + ax + ab \\ x(a + b) = 1 - ab \\ x = (1 - ab) \div (a + b)$

Answer 2

Given


$\frac{1}{abx} = \frac{1}{a } + \frac{1}{b} + \frac{1}{x}$

On the RHS, make the denominators same and add them.

$\frac{1}{a} + \frac{1}{b} + \frac{1}{x} \\ \\ = \frac{1}{a} \times \frac{b}{b} + \frac{1}{b} \times \frac{a}{a} + \frac{1}{x} \\ \\ = \frac{b}{ab} + \frac{a}{ab} + \frac{1}{x} \\ \\ = > \frac{a + b}{ab} + \frac{1}{x} \\ \\ = > \frac{(a + b)}{ab} \times \frac{x}{x} + \frac{1}{x} \times \frac{ab}{ab} \\ \\ = > \frac{x(a + b) + ab}{abx}$


So,

$\frac{1}{abx} = \frac{x(a + b) + ab}{abx}$
cancel abx from both sides as it is the common denominator

$= > 1 = x(a + b) + ab \\ \\ = > x(a + b) = 1 - ab \\ \\ = > x = \frac{1 - ab}{a + b}$

Hope it helps dear friend ☺️