Which of the following is not the solution of
`|2x+5|-|x-3| ge |x+8| ` ?

Options

A. `(-oo,-8]`
B. `[3,oo)`
C. `(-8,3)`
D. none of these

Explanation

To solve the inequality \( |2x + 5| - |x - 3| \geq |x + 8| \), we will break it down step by step.

### Step 1: Understand the Absolute Values
We need to analyze the absolute value expressions involved in the inequality. The critical points where the expressions change are found by setting the inside of the absolute values equal to zero:
- For \( |2x + 5| \): \( 2x + 5 = 0 \) → \( x = -\frac{5}{2} \)
- For \( |x - 3| \): \( x - 3 = 0 \) → \( x = 3 \)
- For \( |x + 8| \): \( x + 8 = 0 \) → \( x = -8 \)

The critical points are \( x = -8, -\frac{5}{2}, 3 \).

### Step 2: Test Intervals
We will test the intervals defined by these critical points:
1. \( (-\infty, -8) \)
2. \( (-8, -\frac{5}{2}) \)
3. \( (-\frac{5}{2}, 3) \)
4. \( (3, \infty) \)

### Step 3: Analyze Each Interval
**Interval 1: \( (-\infty, -8) \)**
- Here, \( 2x + 5 < 0 \), \( x - 3 < 0 \), and \( x + 8 < 0 \).
- The inequality becomes:
\[ -(2x + 5) - (-(x - 3)) \geq -(x + 8) \]
Simplifying gives:
\[ -2x - 5 + x - 3 \geq -x - 8 \]
\[ -x - 8 \geq -x - 8 \quad \text{(True)} \]

**Interval 2: \( (-8, -\frac{5}{2}) \)**
- Here, \( 2x + 5 < 0 \), \( x - 3 < 0 \), and \( x + 8 > 0 \).
- The inequality becomes:
\[ -(2x + 5) - (-(x - 3)) \geq x + 8 \]
Simplifying gives:
\[ -2x - 5 + x - 3 \geq x + 8 \]
\[ -x - 8 \geq x + 8 \quad \text{(False)} \]

**Interval 3: \( (-\frac{5}{2}, 3) \)**
- Here, \( 2x + 5 > 0 \), \( x - 3 < 0 \), and \( x + 8 > 0 \).
- The inequality becomes:
\[ (2x + 5) - (-(x - 3)) \geq x + 8 \]
Simplifying gives:
\[ 2x + 5 + x - 3 \geq x + 8 \]
\[ 2x + 2 \geq x + 8 \]
\[ x \geq 6 \quad \text{(False)} \]

**Interval 4: \( (3, \infty) \)**
- Here, \( 2x + 5 > 0 \), \( x - 3 > 0 \), and \( x + 8 > 0 \).
- The inequality becomes:
\[ (2x + 5) - (x - 3) \geq x + 8 \]
Simplifying gives:
\[ 2x + 5 - x + 3 \geq x + 8 \]
\[ x + 8 \geq x + 8 \quad \text{(True)} \]

### Step 4: Combine Results
From our analysis:
- The inequality holds for intervals \( (-\infty, -8) \) and \( (3, \infty) \).
- The inequality does not hold for intervals \( (-8, -\frac{5}{2}) \) and \( (-\frac{5}{2}, 3) \).

### Conclusion
The solution set for the inequality is:
\[ (-\infty, -8) \cup (3, \infty) \]
Thus, any value in the intervals \( (-8, -\frac{5}{2}) \) or \( (-\frac{5}{2}, 3) \) is not a solution.