Solve -1/x + 1 - m = -1 for x in terms of m.

Step-by-step solution

1. 1

   Record the domain first. The term $\tfrac1x$ requires $x\neq 0$, and since $\tfrac1x$ can never be $0$, the final formula must never evaluate to $0$ either — a useful consistency check later.

2. 2

   Isolate the reciprocal term. Subtract $1$ from both sides, then add $m$:

   $-\frac{1}{x}-m=-2\quad\Longrightarrow\quad -\frac{1}{x}=m-2$

3. 3

   Clear the leading minus sign. Multiply both sides by $-1$:

   $\frac{1}{x}=2-m$

4. 4

   Take reciprocals of both sides. This is legitimate provided neither side is zero:

   $x=\frac{1}{2-m}$

   and it never produces $x=0$, consistent with the domain noted at the start.

5. 5

   Identify the excluded parameter value and check. If $m=2$ the equation becomes $\tfrac1x=0$, which has no solution, so

   $x=\frac{1}{2-m},\qquad m\neq 2$

   Test $m=1$: then $x=1$, and $-1+1-1=-1$ $\checkmark$. Test $m=5$: then $x=-\tfrac13$, and $3+1-5=-1$ $\checkmark$.