Question

Solve 4/x-4=x/x-4-4/3 for x and determine if the solution is extraneous or not

Answer 1

x = 4 (extraneous solution)

Explanation:

`\frac { 4 } { x - 4 } = \frac { x } { x - 4 } - \frac { 4 } { 3 } \\ \frac { 4 } { x - 4 } - \frac { x } { x - 4 } = - \frac { 4 } { 3 } \\ \frac { 4 - x } { x - 4 } = - \frac { 4 } { 3 } \\ 3 ( 4 - x ) = - 4 ( x - 4 ) \\ 1 2 - 3 x = - 4 x + 1 6 \\ 4 x - 3 x = 1 6 - 1 2 \\ x = 4 \\`

This solution is extraneous. Reason being that even if it can be solved algebraically, it is still not a valid solution because if we substitute back
`x=4`, we will get two fractions with zero denominator which would be undefined.