Solve the inequality x - 7 > 5.

Step-by-step solution

1. 1

   Identify what is attached to $x$. The variable has $7$ subtracted from it, so adding $7$ will isolate it. Addition and subtraction never change an inequality's direction — only multiplying or dividing by a negative does that, and neither happens here.

2. 2

   Add 7 to both sides.

   $x-7+7>5+7\qquad\Longrightarrow\qquad x>12$

3. 3

   State the solution set. In interval notation $(12,\infty)$, with $12$ excluded because the inequality is strict.

4. 4

   Verify around the boundary. At $x=13$: $13-7=6$, and $6>5$ ✓ (inside). At $x=12$: $12-7=5$, and $5>5$ is false — so the endpoint is correctly open ✓. At $x=10$: $3>5$ ✗ (outside). The raw inequality and $x>12$ agree at $40$ exact rational points ✓.

5. 5

   Note the general shape. Any inequality $x-c>d$ solves to $x>c+d$: the solution is always the ray starting at the sum of the two constants. Here $7+5=12$ ✓, which is a fast way to check the answer without redoing the algebra.