How do you solve for x: $x/(x-3) = 3/(x-3) + 3$ ?

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Answer 1 · 2016-08-04T11:37:27

$x in O/$

The first thing to do here is get rid of the denominators. Note that a possible solution to the original equation must satisfy the condition

$color(purple)(|bar(ul(color(white)(a/a)color(black)(x - 3 != 0 implies x != 3)color(white)(a/a)|)))$

So, to get rid of the denominators, multiply the last term by $1 = (x-3)/(x-3)$ . This will get you

$x/(x-3) = 3/(x-3) + 3 * (x-3)/(x-3)$

You can now focus exclusively on the numerators

$x = 3 + 3 * (x-3)$

Rearrange to get $x$ alone on one side of the equation

$x = 3 + 3x - 9$

$x = 3x - 6$

$2x = 6 implies x = 6/2 = 3$

Notice that $x$ came out to be equal to the one value that it cannot take, i.e. $x=3$ .

This means that your original equation has no solution, $x in O/$ .

How do you solve for x: x/(x-3) = 3/(x-3) + 3 x/(x-3) = 3/(x-3) + 3 ?