$$ \left\vert x-1\right\vert+\left\vert x-3\right\vert=\left\vert2x-4\right\vert $$

We need to solve the equation involving absolute values:

\[
\left\vert x - 1 \right\vert + \left\vert x - 3 \right\vert = \left\vert 2x - 4 \right\vert.
\]

### Step 1: Analyze Absolute Values

Break the equation into different cases based on the critical points where the expressions inside the absolute values change sign. These critical points occur at $ x = 1 $, $ x = 3 $, and $ x = 2 $ (since $ |2x - 4| = 2|x - 2| $).

Thus, the three cases we consider are:

1. **Case 1**: $ x < 1 $
2. **Case 2**: $ 1 \leq x \leq 3 $
3. **Case 3**: $ x > 3 $

---

### Case 1: $ x < 1 $
For $ x < 1 $, we have:
\[
\left\vert x - 1 \right\vert = -(x - 1) = 1 - x,
\]
\[
\left\vert x - 3 \right\vert = -(x - 3) = 3 - x,
\]
\[
\left\vert 2x - 4 \right\vert = -(2x - 4) = 4 - 2x.
\]

Substitute these into the equation:
\[
(1 - x) + (3 - x) = 4 - 2x,
\]
\[
4 - 2x = 4 - 2x.
\]

This is an identity, meaning it is true for all $ x $ in the range $ x < 1 $.

Thus, **all $ x < 1 $ are solutions** for this case.

---

### Case 2: $ 1 \leq x \leq 3 $
For $ 1 \leq x \leq 3 $, we have:
\[
\left\vert x - 1 \right\vert = x - 1,
\]
\[
\left\vert x - 3 \right\vert = 3 - x,
\]
\[
\left\vert 2x - 4 \right\vert = 2x - 4.
\]

Substitute these into the equation:
\[
(x - 1) + (3 - x) = 2x - 4,
\]
\[
2 = 2x - 4.
\]

Solving for $ x $:
\[
2x = 6,
\]
\[
x = 3.
\]

Since $ x = 3 $ lies within the interval $ 1 \leq x \leq 3 $, it is a valid solution.

---

### Case 3: $ x > 3 $
For $ x > 3 $, we have:
\[
\left\vert x - 1 \right\vert = x - 1,
\]
\[
\left\vert x - 3 \right\vert = x - 3,
\]
\[
\left\vert 2x - 4 \right\vert = 2x - 4.
\]

Substitute these into the equation:
\[
(x - 1) + (x - 3) = 2x - 4,
\]
\[
2x - 4 = 2x - 4.
\]

This is again an identity, meaning it is true for all $ x > 3 $.

Thus, **all $ x > 3 $ are solutions** for this case.

---

### Final Solution

The solution set is:
\[
(-\infty, 1) \cup [3, \infty).
\]

Would you like further details or explanations on any specific part of this solution?

---

Here are 5 related questions to expand on this problem:
1. How do you solve inequalities involving absolute values?
2. Can absolute value equations have no solution? How would you detect that?
3. What are some real-world applications of absolute value equations?
4. How does the graph of an absolute value function look, and how can we interpret it?
5. How can you solve systems of equations that involve absolute values?

**Tip**: When solving absolute value equations, always consider breaking the equation into cases based on where the expressions inside the absolute values change sign.

$$ \left\vert x - 1 \right\vert + \left\vert x - 3 \right\vert = \left\vert 2x - 4 \right\vert $$

Learn how to solve the absolute value equation |x - 1| + |x - 3| = |2x - 4| through step-by-step breakdowns. This solution involves analyzing multiple cases based on critical points, covering key algebraic concepts such as absolute value equations and piecewise functions. Suitable for students in Grades 9-11.

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