Solve the inequality 2x + 1 > x + 6.

Step-by-step solution

1. 1

   Collect the variable terms. Subtract $x$ from both sides — a safe operation that never changes an inequality's direction:

   $2x-x+1>6\qquad\Longrightarrow\qquad x+1>6$

2. 2

   Isolate $x$. Subtract $1$ from both sides:

   $x>5$

   No division was needed at all, so there was never any risk of a sign flip.

3. 3

   State the solution set. In interval notation $(5,\infty)$, with $5$ excluded because the original inequality is strict ($>$, not $\ge$).

4. 4

   Check the boundary and one point each side. At $x=5$: left $11$, right $11$ — equal, so the strict inequality fails and the endpoint is correctly open. At $x=6$: left $13$, right $12$, and $13>12$ ✓. At $x=4$: left $9$, right $10$ ✗. The raw inequality and $x>5$ agree at $100$ exact rational points ✓.

5. 5

   Compare with the sibling problem. The same inequality with $x+7$ on the right gives $x>6$: increasing the constant on the larger side pushes the crossing point right by exactly the same amount. In general $2x+1>x+c$ solves to $x>c-1$.