solve for x
`|x+1| + |x-2| + |x-5| = 4`

Options

A. `-3,1`
B. `2`
C. `-1,1`
D. no solution

Answer

To solve the equation \( |x+1| + |x-2| + |x-5| = 4 \), we need to analyze the absolute value expressions and break the problem into different cases based on the critical points where the expressions inside the absolute values change sign. The critical points are \( x = -1 \), \( x = 2 \), and \( x = 5 \).

### Step 1: Identify the cases based on critical points
We will consider the following intervals:
1. \( x < -1 \)
2. \( -1 \leq x < 2 \)
3. \( 2 \leq x < 5 \)
4. \( x \geq 5 \)

### Step 2: Case 1: \( x < -1 \)
In this case, all expressions inside the absolute values are negative:
- \( |x+1| = -(x+1) = -x - 1 \)
- \( |x-2| = -(x-2) = -x + 2 \)
- \( |x-5| = -(x-5) = -x + 5 \)

Substituting these into the equation:
\[ -x - 1 - x + 2 - x + 5 = 4 \]
Simplifying:
\[ -3x + 6 = 4 \]
\[ -3x = 4 - 6 \]
\[ -3x = -2 \implies x = \frac{2}{3} \]
Since \( \frac{2}{3} \) is not in the interval \( x < -1 \), there are no solutions in this case.

### Step 3: Case 2: \( -1 \leq x < 2 \)
In this case:
- \( |x+1| = x + 1 \)
- \( |x-2| = -(x-2) = -x + 2 \)
- \( |x-5| = -(x-5) = -x + 5 \)

Substituting these into the equation:
\[ x + 1 - x + 2 - x + 5 = 4 \]
Simplifying:
\[ 8 - x = 4 \]
\[ -x = 4 - 8 \]
\[ -x = -4 \implies x = 4 \]
Since \( 4 \) is not in the interval \( -1 \leq x < 2 \), there are no solutions in this case.

### Step 4: Case 3: \( 2 \leq x < 5 \)
In this case:
- \( |x+1| = x + 1 \)
- \( |x-2| = x - 2 \)
- \( |x-5| = -(x-5) = -x + 5 \)

Substituting these into the equation:
\[ x + 1 + x - 2 - x + 5 = 4 \]
Simplifying:
\[ x + 4 = 4 \]
\[ x = 0 \]
Since \( 0 \) is not in the interval \( 2 \leq x < 5 \), there are no solutions in this case.

### Step 5: Case 4: \( x \geq 5 \)
In this case, all expressions inside the absolute values are positive:
- \( |x+1| = x + 1 \)
- \( |x-2| = x - 2 \)
- \( |x-5| = x - 5 \)

Substituting these into the equation:
\[ x + 1 + x - 2 + x - 5 = 4 \]
Simplifying:
\[ 3x - 6 = 4 \]
\[ 3x = 10 \]
\[ x = \frac{10}{3} \approx 3.33 \]
Since \( \frac{10}{3} \) is not in the interval \( x \geq 5 \), there are no solutions in this case.

### Conclusion
After analyzing all cases, we find that there are no values of \( x \) that satisfy the equation \( |x+1| + |x-2| + |x-5| = 4 \). Therefore, the solution is:

**No solution.**