Solve the inequality 7|w - 6| >= 2.

Step-by-step solution

1. 1

   Isolate the absolute value before splitting anything. The rewrite rules for $|A|$ only apply when the absolute value stands alone. Here it is multiplied by $7$, so divide both sides by the positive number $7$ (no sign flip):

   $|w-6|\ge\frac{2}{7}$

2. 2

   Split a greater-than-or-equal absolute value into two rays. $|A|\ge b$ with $b>0$ says the distance from $A$ to $0$ is at least $b$, which happens on the two sides away from the origin — not between them:

   $w-6\le-\frac{2}{7}\qquad\text{or}\qquad w-6\ge\frac{2}{7}$

   The word is or, never and: no single $w$ can be both far left and far right at once.

3. 3

   Solve each branch by adding 6. Convert $6$ to sevenths so the fractions combine cleanly, $6=\tfrac{42}{7}$:

   $w\le\frac{42}{7}-\frac{2}{7}=\frac{40}{7}\qquad\text{or}\qquad w\ge\frac{42}{7}+\frac{2}{7}=\frac{44}{7}$

4. 4

   Write the solution set. In interval notation

   $\left(-\infty,\tfrac{40}{7}\right]\cup\left[\tfrac{44}{7},\infty\right)$

   Both endpoints are included because the original relation allowed equality. Numerically the excluded gap is the short open interval $(5.714\ldots,\,6.285\ldots)$ centred on $w=6$.

5. 5

   Verify the boundary and the gap. At $w=\tfrac{40}{7}$: $7\left|\tfrac{40}{7}-6\right|=7\cdot\tfrac{2}{7}=2\ge2$ ✓ (equality, so the endpoint belongs). At the centre $w=6$: $7|0|=0$, which is not $\ge2$ ✗, confirming the gap. Comparing the raw inequality with the claimed set at $500$ exact rational points agrees everywhere ✓.