Answer :
We start with the inequality
$$
|x-2| + 3 > 17.
$$
**Step 1. Isolate the absolute value.**
Subtract 3 from both sides:
$$
|x-2| > 17 - 3,
$$
which simplifies to
$$
|x-2| > 14.
$$
**Step 2. Split the absolute value inequality into two cases.**
Since an absolute value inequality of the form
$$
|A| > B
$$
(where $B > 0$) is equivalent to
$$
A > B \quad \text{or} \quad A < -B,
$$
for our inequality the two cases are:
1. $x - 2 > 14$, and
2. $x - 2 < -14$.
**Step 3. Solve both cases.**
1. For $x - 2 > 14$:
Add 2 to both sides:
$$
x > 14 + 2,
$$
so
$$
x > 16.
$$
2. For $x - 2 < -14$:
Add 2 to both sides:
$$
x < -14 + 2,
$$
so
$$
x < -12.
$$
**Step 4. Combine the solutions.**
The inequality holds true when either
$$
x < -12 \quad \text{or} \quad x > 16.
$$
Thus, the solution to the inequality is
$$
\boxed{x < -12 \quad \text{or} \quad x > 16.}
$$
$$
|x-2| + 3 > 17.
$$
**Step 1. Isolate the absolute value.**
Subtract 3 from both sides:
$$
|x-2| > 17 - 3,
$$
which simplifies to
$$
|x-2| > 14.
$$
**Step 2. Split the absolute value inequality into two cases.**
Since an absolute value inequality of the form
$$
|A| > B
$$
(where $B > 0$) is equivalent to
$$
A > B \quad \text{or} \quad A < -B,
$$
for our inequality the two cases are:
1. $x - 2 > 14$, and
2. $x - 2 < -14$.
**Step 3. Solve both cases.**
1. For $x - 2 > 14$:
Add 2 to both sides:
$$
x > 14 + 2,
$$
so
$$
x > 16.
$$
2. For $x - 2 < -14$:
Add 2 to both sides:
$$
x < -14 + 2,
$$
so
$$
x < -12.
$$
**Step 4. Combine the solutions.**
The inequality holds true when either
$$
x < -12 \quad \text{or} \quad x > 16.
$$
Thus, the solution to the inequality is
$$
\boxed{x < -12 \quad \text{or} \quad x > 16.}
$$
