Question

How do you solve `1/(x-2) = 3/4 - 1/x`?

Answer 1

You must put on an equal denominator.

Explanation:

The LCD (Least Common Denominator) is `4(x)(x - 2)`

`(1(4x))/((x - 2)(x)) = (3(x)(x - 2))/(4(x)(x - 2)) - (4x - 8)/(x(4x - 8))`

Since everything is now equivalent, we can eliminate the denominators and solve this equation as a simple quadratic.

`4x = 3x^2 - 6x - 4x + 8`

`0 = 3x^2 - 14x + 8`

We can solve by factoring. To factor a trinomial of the form `y = ax^2 + bx + c, a != 1`, you must find a pair of numbers that multiply to `a xx c` and that add to b. Two numbers that multiply to `24` and that add to `-14` are `-12 and -2`

`0 = 3x^2 - 12x - 2x + 8`

`0 = 3x(x - 4) - 2(x - 4)`

`0 = (3x - 2)(x - 4)`

`x = 2/3 and 4`

Checking both solutions we find that both satisfy the original equation. Thus, our solution set is `{x = 2/3 and 4}`

Hopefully this helps!