Question

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Solve the following equation. Show all your work.
(x/(x−2))+((x−1)/(x+1))=−1

Answer 1

`\displaystyle x = 0 \text{ and } x = 1`

Explanation:

We are given and asked to solve the equation:

`\displaystyle (x)/(x-2) + (x-1)/(x+1) = -1`

We can first remove the rational expressions by multiplying both sides of the equation by its LCM (in this case, by (x - 2)(x + 1)):

`\displaystyle (x-2)(x+1) \left( \displaystyle (x)/(x-2) + (x-1)/(x+1)\right) = -1(x-2)(x+1)`

Simplify:

`\displaystyle x(x+1)+(x-1)(x-2) = -(x-2)(x+1)`

Expand and isolate:

`\displaystyle \begin{aligned} (x^2 + x) + (x^2 - 3x + 2) & = -(x^2 -x -2) \\ \\ 2x^2 - 2x + 2& = -x^2 + x + 2 \\ \\ 3x^2 - 3x & = 0 \end{aligned}`

Hence solve for x:

`\displaystyle \begin{aligned} 3x(x - 1) & = 0 \\ \\ x & = 0\text{ or } x = 1\end{aligned}`

In conclusion, the solutions of the equation are x = 0 and x = 1.