solve: `|x-2|+|x-4| ge 8`

Explanation

To solve the inequality \( |x-2| + |x-4| \geq 8 \), we will analyze the expression by breaking it down into intervals based on the critical points where the absolute values change, which are \( x = 2 \) and \( x = 4 \).

### Step 1: Identify critical points
The critical points are found by setting the expressions inside the absolute values to zero:
- \( x - 2 = 0 \) gives \( x = 2 \)
- \( x - 4 = 0 \) gives \( x = 4 \)

These points divide the number line into three intervals:
1. \( (-\infty, 2) \)
2. \( [2, 4] \)
3. \( (4, \infty) \)

### Step 2: Analyze the first interval \( (-\infty, 2) \)
In this interval, both \( x - 2 \) and \( x - 4 \) are negative. Therefore, we can rewrite the absolute values:
\[ |x-2| = -(x-2) = -x + 2 \]
\[ |x-4| = -(x-4) = -x + 4 \]
Now, substitute these into the inequality:
\[ -x + 2 - x + 4 \geq 8 \]
Combine like terms:
\[ -2x + 6 \geq 8 \]
Subtract 6 from both sides:
\[ -2x \geq 2 \]
Divide by -2 (remember to reverse the inequality):
\[ x \leq -1 \]
Since we are in the interval \( (-\infty, 2) \), the solution here is:
\[ x \in (-\infty, -1] \]

### Step 3: Analyze the second interval \( [2, 4] \)
In this interval, \( x - 2 \) is non-negative and \( x - 4 \) is non-positive. Thus:
\[ |x-2| = x - 2 \]
\[ |x-4| = -(x-4) = -x + 4 \]
Substituting into the inequality:
\[ x - 2 - x + 4 \geq 8 \]
Combine like terms:
\[ 2 \geq 8 \]
This statement is false, so there are no solutions in this interval.

### Step 4: Analyze the third interval \( (4, \infty) \)
In this interval, both \( x - 2 \) and \( x - 4 \) are positive:
\[ |x-2| = x - 2 \]
\[ |x-4| = x - 4 \]
Substituting into the inequality:
\[ x - 2 + x - 4 \geq 8 \]
Combine like terms:
\[ 2x - 6 \geq 8 \]
Add 6 to both sides:
\[ 2x \geq 14 \]
Divide by 2:
\[ x \geq 7 \]
Since we are in the interval \( (4, \infty) \), the solution here is:
\[ x \in [7, \infty) \]

### Final Solution
Combining the solutions from all intervals, we get:
\[ x \in (-\infty, -1] \cup [7, \infty) \]