Question

How do you solve `(x+4) /( x-2)=12`?

Answer 1

graph{0x+12 [-9.74, 10.26, 6.4, 16.4]}
graph{(x+4)/(x-2) [-25.08, 26.24, -10.77, 14.9]}

You can solve it in two ways, first being algebraically.
`x+4=12(x-2)`
`x+4=12x-24`
`4=11x-24`
`11x=28`
`x=28/11`

You can also solve it graphically. Plot the graphs of `y=12` and `y=(x+4)(x-2)` and find out where the two lines cross. The x-coordinate is your answer.

Answer 2

Multiply both sides of the equation by the denominator to 'move' `x-2` out of the denominator position and solve to find that `x=28//11`.

Explanation:

In a situation like this, it is easier to multiply both sides of an equation by the denominator of the fraction to 'clear' the denominator.

Multiplying both sides keeps the expression 'even' since we're increasing both sides by the same multiple.

`(x+4)/(x-2) xx (x-2)=12xx(x-2)`

Now that we've multiplied, we can rearrange the fraction slightly:

`(x+4)xxcancel((x-2)/(x-2))1=12xx(x-2)`

`x+4=12xx(x-2)`

`x+4=12x-24`

Now that we've 'cleared' the denominator, we can solve for `x` easily. first, we'll subtract `x` from both sides, to 'move' all `x`-terms to the Right Hand Side (RHS):

`cancel(x)+4cancel(color(red)(-x))=12x-24 color(red)(-x)`

`4=11x-24`

Next, we'll add 24 to both sides, to get all of the constants onto the Left Hand Side (LHS):

`4color(red)(+24)=11xcancel(-24color(red)(+24))`

`28=11x`

Finally, divide both sides by 11 to solve for `x`:

`28/color(red)(11)=(cancel(11)x)/cancel(color(red)(11))`

`color(green)(rArr x=28/11=28//11)`