Solve the inequality 2x + 1 > x + 7.

Step-by-step solution

1. 1

   Gather the variable on the left. Subtracting the same expression from both sides is always safe for an inequality — the direction is unaffected. Subtract $x$:

   $2x-x+1>7\qquad\Longrightarrow\qquad x+1>7$

   Because $2x-x=x$ has a positive coefficient, no division by a negative will be needed later.

2. 2

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

   $x>6$

3. 3

   State the solution set. In interval notation $(6,\infty)$, with $6$ excluded because the inequality is strict.

4. 4

   Verify on both sides of the boundary. At $x=7$: left $15$, right $14$, and $15>14$ ✓. At $x=6$: left $13$, right $13$ — equal, so the strict inequality fails and the endpoint is correctly left out. At $x=0$: left $1$, right $7$, and $1>7$ ✗ ✓ (correctly outside). Comparing the raw inequality with $x>6$ at $100$ exact rational points agrees everywhere ✓.

5. 5

   See it as two lines. $y=2x+1$ has slope $2$ and $y=x+7$ has slope $1$; the steeper line overtakes the shallower one at their crossing point $x=6$ and stays above it thereafter. Different slopes guarantee exactly one crossing, so the answer must be a single ray.