The number of integers satisfying `|2x-3|+|x+5| le |x-8|` is

Options

A. `5`
B. `6`
C. `7`
D. `8`

Explanation

To solve the inequality \( |2x - 3| + |x + 5| \leq |x - 8| \), we will follow these steps:

### Step 1: Identify Critical Points
The critical points occur where the expressions inside the absolute values are zero. We set each expression to zero:

1. \( 2x - 3 = 0 \) → \( x = \frac{3}{2} \)
2. \( x + 5 = 0 \) → \( x = -5 \)
3. \( x - 8 = 0 \) → \( x = 8 \)

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

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

### Step 3: Analyze Each Interval

**Interval 1: \( (-\infty, -5) \)**
- Choose \( x = -6 \):
\[ |2(-6) - 3| + |-6 + 5| = | -12 - 3 | + | -1 | = | -15 | + | -1 | = 15 + 1 = 16 \]
\[ | -6 - 8 | = | -14 | = 14 \]
Thus, \( 16 \not\leq 14 \) (not satisfied).

**Interval 2: \( (-5, \frac{3}{2}) \)**
- Choose \( x = 0 \):
\[ |2(0) - 3| + |0 + 5| = | -3 | + | 5 | = 3 + 5 = 8 \]
\[ | 0 - 8 | = | -8 | = 8 \]
Thus, \( 8 \leq 8 \) (satisfied).

**Interval 3: \( (\frac{3}{2}, 8) \)**
- Choose \( x = 2 \):
\[ |2(2) - 3| + |2 + 5| = | 4 - 3 | + | 7 | = 1 + 7 = 8 \]
\[ | 2 - 8 | = | -6 | = 6 \]
Thus, \( 8 \not\leq 6 \) (not satisfied).

**Interval 4: \( (8, \infty) \)**
- Choose \( x = 9 \):
\[ |2(9) - 3| + |9 + 5| = | 18 - 3 | + | 14 | = 15 + 14 = 29 \]
\[ | 9 - 8 | = | 1 | = 1 \]
Thus, \( 29 \not\leq 1 \) (not satisfied).

### Step 4: Determine Valid Intervals
From our testing, the only interval that satisfies the inequality is \( (-5, \frac{3}{2}) \).

### Step 5: Find Integer Solutions
Now we find the integers in the interval \( (-5, \frac{3}{2}) \):
- The integers are: \( -4, -3, -2, -1, 0, 1 \).

### Conclusion
The total number of integers satisfying the inequality is **6**.