Question

2. Find all numbers \( x \) such that \[ \frac{x+2}{x+1}=\frac{1}{x-1}+2 \]

Answer 1

Explanation:

To find all numbers x that satisfy the equation:

\frac{x+2}{x+1} = \frac{1}{x-1} + 2

x+1

x+2

​

=

x−1

1

​

+2

We can simplify and solve the equation step by step:

Step 1: Multiply both sides of the equation by (x+1)(x-1)(x+1)(x−1) to clear the fractions:

(x+2)(x-1) = (x+1) + 2(x-1)(x+2)(x−1)=(x+1)+2(x−1)

Step 2: Expand and simplify:

x^2 + x - 2 = x + 1 + 2x - 2x

2

+x−2=x+1+2x−2

x^2 + x - 2 = 3x - 1x

2

+x−2=3x−1