We are tasked with solving the inequality:

\[
|x - 10| < 3|2 - x|
\]

### Step 1: Consider two cases for the absolute value of \( |x - 10| \) and \( |2 - x| \).

For absolute value expressions, we need to break the equation into different cases depending on whether the expressions inside the absolute values are positive or negative. 

### Case 1: \( x \geq 10 \)

Here, both expressions inside the absolute values are non-negative, so we can remove the absolute value symbols directly:

\[
x - 10 < 3(2 - x)
\]

Expanding the right-hand side:

\[
x - 10 < 6 - 3x
\]

Now, let's solve for \( x \):

\[
x + 3x < 6 + 10
\]
\[
4x < 16
\]
\[
x < 4
\]

But in this case, we assumed \( x \geq 10 \), and since \( x < 4 \) contradicts this, **there is no solution in this case**.

### Case 2: \( x < 10 \)

Now, we need to consider two further subcases: \( x \geq 2 \) and \( x < 2 \).

#### Subcase 2.1: \( x \geq 2 \)

For \( x \geq 2 \), \( x - 10 \) is negative and \( 2 - x \) is non-negative. So:

\[
-(x - 10) < 3(2 - x)
\]

This simplifies to:

\[
- x + 10 < 6 - 3x
\]

Solving for \( x \):

\[
- x + 3x < 6 - 10
\]
\[
2x < -4
\]
\[
x < -2
\]

But we assumed \( x \geq 2 \), so there is no solution in this subcase.

#### Subcase 2.2: \( x < 2 \)

For \( x < 2 \), both expressions inside the absolute values are negative. Hence:

\[
-(x - 10) < 3(-(2 - x))
\]

Simplifying:

\[
- x + 10 < 3(x - 2)
\]
\[
- x + 10 < 3x - 6
\]

Now solve for \( x \):

\[
- x - 3x < -6 - 10
\]
\[
- 4x < -16
\]
\[
x > 4
\]

But we assumed \( x < 2 \), so there is no solution in this subcase either.

### Conclusion:

The inequality \( |x - 10| < 3|2 - x| \) has **no solution** because in every case, the conditions contradict the assumed ranges of \( x \).

---

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the general method for solving absolute value inequalities?
2. How would the solution change if one side of the inequality were just a constant?
3. Can you apply this method to other inequalities involving absolute values?
4. What happens if we change the inequality from "<" to "≤"?
5. How can graphing help in understanding absolute value inequalities?

**Tip:** When dealing with absolute value inequalities, always consider different cases based on the sign of the expressions inside the absolute value functions!

Learn how to solve the absolute value inequality |x-10|<3|2-x| with a detailed step-by-step solution. This problem involves analyzing cases based on the values of x and breaking down the inequality into different scenarios. Suitable for students in Grades 9-12.

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