Solve the inequality -x + 1 > 0.

Step-by-step solution

1. 1

   Isolate the term containing $x$. Subtract $1$ from both sides — subtraction never disturbs the direction:

   $-x>-1$

2. 2

   Divide by $-1$ and flip the sign. This is the one rule that governs the problem: multiplying or dividing an inequality by a negative number reverses its direction:

   $x<1$

   Keeping $>$ would give the exactly wrong answer $x>1$. The reason is concrete: $-2<-1$ but $2>1$ — negating swaps the order of every pair of numbers.

3. 3

   Cross-check by an alternative route that avoids the flip. From $-x+1>0$, add $x$ to both sides instead:

   $1>x$

   which reads as $x<1$ ✓ — the same answer with no negative division at all. When in doubt, moving the variable to the side where its coefficient is positive sidesteps the rule entirely.

4. 4

   State the solution set. In interval notation $(-\infty,1)$, with $1$ excluded since the inequality is strict.

5. 5

   Verify around the boundary. At $x=0$: $-0+1=1>0$ ✓ (inside). At $x=1$: $-1+1=0$, and $0>0$ is false — endpoint correctly open ✓. At $x=2$: $-2+1=-1>0$ ✗ (outside). The raw inequality and $x<1$ agree at $40$ exact rational points ✓.