Question

How do you solve `6 / (x-4) + 9/x = -36/(x^2 - 4x)`?

Answer 1

`x="no solution"`

Explanation:

Before starting, it is important to note the restrictions in this equation. When the denominator of each fraction is set to not equal `0`, the restrictions are:

`x-4!=0color(white)(XXXXX)x!=0color(white)(XXXXX)x^2-4x!=0`

`x!=4color(white)(XXXXXXXXXXXXXXX)x(x-4)!=0`

`color(white)(XXXXXXXXXXXXXXXXXX)x!=0,4`

Thus, the restrictions are `color(red)(|bar(ul(color(white)(a/a)x!=0,4color(white)(a/a)|)))`.

Solving the Equation
`1`. Start by adding the two fractions on the left side of the equation.

`6/(x-4)+9/x=-36/(x^2-4x)`

`(6color(orange)x)/(color(orange)x(x-4))+(9color(blue)((x-4)))/(xcolor(blue)((x-4)))=-36/(x^2-4x)`

`2`. Simplify.

`(6x)/(x^2-4x)+(9x-36)/(x^2-4x)=-36/(x^2-4x)`

`(15x-36)/(x^2-4x)=-36/(x^2-4x)`

`3`. Multiply both sides by `color(purple)(x^2-4x)` to get rid of the denominators.

`color(purple)((x^2-4x))((15x-36)/(x^2-4x))=color(purple)((x^2-4x))(-36/(x^2-4x))`

`4`. Simplify.

`color(red)cancelcolor(purple)((x^2-4x))((15x-36)/color(red)cancelcolor(black)((x^2-4x)))=color(red)cancelcolor(purple)((x^2-4x))(-36/color(red)cancelcolor(black)((x^2-4x)))`

`15x-36=-36`

`5`. Solve for `x`.

`15x=0`

`x=0`

However, looking back at the restrictions `(color(red)(x!=0,4))`, `x=0` is not a valid solution. Therefore:

`color(green)(|bar(ul(color(white)(a/a)x="no solution"color(white)(a/a)|)))`