QuestionSolve for all values of $x$ : $\frac{1}{x-1}-2=\frac{x}{x-1}$

Studdy Solution

STEP 1

What is this asking? We need to find all the $x$ values that make this funky equation with fractions true! Watch out! Remember, we can't divide by zero, so keep an eye on that denominator and make sure $x-1$ isn't ever zero!

STEP 2

1. Combine the fractions
2. Multiply both sides
3. Solve for $x$
4. Check for validity

STEP 3

Alright, let's tackle those fractions!
Since both fractions have the same denominator, $(x-1)$ , we can combine them directly.
Remember, it's like adding slices of the same size pie!
We have $\frac{1}{x-1}$ and we're subtracting $\frac{x}{x-1}$ , which gives us $\frac{1-x}{x-1}$ .

STEP 4

So, our equation now looks like this: $\frac{1-x}{x-1} - 2 = 0$

STEP 5

Let's get rid of that fraction!
We can do this by multiplying both sides of the equation by $(x-1)$ .
Remember, whatever we do to one side, we gotta do to the other to keep things balanced.
On the left side, the $(x-1)$ in the numerator and denominator divide to one, leaving us with $1-x$ .
On the right side, $0 \cdot (x-1)$ is just $0$ .

STEP 6

Our equation is now: $1 - x - 2 \cdot (x-1) = 0$

STEP 7

Time to unwrap that $x$ !
First, let's distribute that $-2$ to both terms inside the parentheses: $-2 \cdot x$ gives us $-2x$ , and $-2 \cdot -1$ gives us $+2$ .

STEP 8

Now, our equation looks like this: $1 - x - 2x + 2 = 0$

STEP 9

Let's combine like terms.
We've got a $-x$ and a $-2x$ , which combine to $-3x$ .
And we have a $1$ and a $2$ , which combine to $3$ .

STEP 10

So, our simplified equation is: $3 - 3x = 0$

STEP 11

Now, let's isolate $x$ .
We can subtract $3$ from both sides: $-3x = -3$

STEP 12

Finally, divide both sides by $-3$ to get $x$ all by itself: $x = \frac{-3}{-3} = 1$

STEP 13

Hold on!
Remember back in our first observation, we said $x$ can't be $1$ , because that would make the denominator zero.
And guess what?
We found $x=1$ !
So, even though we did all that awesome math, this solution isn't valid.

STEP 14

There's no value of $x$ that satisfies the original equation.
Our final answer is: no solution!

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