Question

How do you solve `\frac { 4} { x - 2} = \frac { - 3} { x + 5} + \frac { 7} { ( x + 5) ( x - 2) }`?

Answer 1

Please see the explanation.

Explanation:

Begin by stipulating that `x = 2` and `x = -5` are invalid solutions for the equation, because they would cause division by zero.

`4/(x - 2) = -3/(x + 5) + 7/((x + 5)(x - 2)); x !=2 and x!=-5`

Move the second term to the left side:

`4/(x - 2) + 3/(x + 5) = 7/((x + 5)(x - 2)); x !=2 and x!=-5`

Remove all of the denominators by multiply both sides by `((x + 5)(x - 2))`

`4(x + 5) + 3(x - 2) = 7;x !=2 and x!=-5`

Use the distributive property where appropriate:

`4x + 20 + 3x - 6 = 7;x !=2 and x!=-5`

Subtract 14 from both sides:

`4x + 3x = -7;x !=2 and x!=-5`

Combine like terms:

`7x = -7;x !=2 and x!=-5`

`x = -1` and we can drop the restrictions.

check:

`4/(-1 - 2) = -3/(-1 + 5) + 7/((-1 + 5)(-1 - 2))`

`-4/3 = -3/4 + 7/-12`

`-4/3 = -4/3`

This checks