Question

How do you solve `\frac { 3} { x + 2} + \frac { 7} { x } = \frac { 8} { x ^ { 2} + 2x }`?

Answer 1

`x=-3/5`

Explanation:

`3/(x+2) +7/x= 8/(x^2+2x)`

Factor out the GCF `x` from the denominator in the term on the right.

`3/(x+2) +7/x= 8/(x(x+2))`

Get a common denominator.

`x/x * 3/(x+2)+ ((x+2))/((x+2)) * 7/x = 8/(x(x+2))`

Now that all the denominator are the same, they can be "removed" from the equation and it can be solved using just the numerators.

`3x +7(x+2)=8`

`3x+7x+14=8`

`10x+14=color(white)(aa)8`
`color(white)(aaa)-14color(white)(a^2)-14`

`10x=-6`

`(10x)/10=( -6)/10`

`x= -3/5`

Note that solutions to rational equations must be "checked" to make sure they do not result in dividing by zero. In this example, the only values of `x` that would result in a denominator of zero would be `x=0` or `x=-2`, so `x= -3/5` is valid.