Solve the inequality x/2 > x - 1.

Step-by-step solution

1. 1

   Clear the fraction by multiplying by 2. The multiplier $2$ is positive, so the inequality direction is unchanged — this is safe in a way that multiplying by a variable would not be:

   $x>2(x-1)=2x-2$

   Note the $2$ must reach both terms of $x-1$.

2. 2

   Collect the variable terms. Subtract $2x$ from both sides:

   $x-2x>-2\qquad\Longrightarrow\qquad-x>-2$

   The coefficient is now negative, which sets up the one step that requires care.

3. 3

   Multiply by $-1$ and reverse the inequality. Negating both sides of an inequality flips its direction:

   $x<2$

   Skipping the flip would give the exactly wrong answer $x>2$, so this is the crux of the problem. The alternative that avoids it entirely: from $x>2x-2$, add $2$ and subtract $x$ to get $2>x$ directly, then read it backwards as $x<2$ ✓ — same answer, no negative coefficient.

4. 4

   State the solution set. In interval notation $(-\infty,2)$, with $2$ excluded because the relation is strict.

5. 5

   Verify around the boundary. At $x=0$: left $0$, right $-1$, and $0>-1$ ✓ (inside). At $x=4$: left $2$, right $3$, and $2>3$ ✗ (outside). At $x=2$: left $1$, right $1$ — equal, so the strict inequality fails and the endpoint is correctly open ✓. The raw inequality and $x<2$ agree at $80$ exact rational points ✓.