Solve the inequality 7|w - 6| >= 21.

Step-by-step solution

1. 1

   Divide by the positive coefficient. The bars must be alone before splitting, and dividing by $7>0$ leaves the direction unchanged:

   $|w-6|\ge 3$

2. 2

   Read the inequality as a distance statement. $|w-6|$ is the distance from $w$ to $6$, so the inequality says: $w$ is at least $3$ units away from $6$. Walking $3$ units left of $6$ lands on $3$; walking $3$ units right lands on $9$.

3. 3

   Write the two branches formally. For $b>0$, $|A|\ge b$ becomes $A\le -b$ or $A\ge b$:

   $w-6\le -3\qquad\text{or}\qquad w-6\ge 3$

4. 4

   Add $6$ to each branch. No sign flipping is needed because adding a constant never changes direction:

   $w\le 3\qquad\text{or}\qquad w\ge 9$

5. 5

   Confirm the endpoints belong. The inequality is non-strict, so test them: $7|3-6|=21\ge 21$ $\checkmark$ and $7|9-6|=21\ge 21$ $\checkmark$. A midpoint check rules out the middle: $7|6-6|=0\not\ge 21$ $\checkmark$. The solution set is $(-\infty,3]\cup[9,\infty)$.