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 break it down step by step.

### Step 1: Identify the critical points
The expressions inside the absolute values will change at certain values of \( x \). We need to find these critical points:
- For \( |2x - 3| \), it changes at \( 2x - 3 = 0 \) or \( x = \frac{3}{2} \).
- For \( |x + 5| \), it changes at \( x + 5 = 0 \) or \( x = -5 \).
- For \( |x - 8| \), it changes at \( x - 8 = 0 \) or \( x = 8 \).

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

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

### Step 3: Analyze each interval

**Interval 1: \( (-\infty, -5) \)**
- Choose \( x = -6 \):
\[ |2(-6) - 3| + |-6 + 5| = | -12 - 3 | + | -1 | = 15 + 1 = 16 \]
\[ | -6 - 8 | = | -14 | = 14 \]
Since \( 16 \nleq 14 \), this interval does not satisfy the inequality.

**Interval 2: \( [-5, \frac{3}{2}) \)**
- Choose \( x = 0 \):
\[ |2(0) - 3| + |0 + 5| = | -3 | + | 5 | = 3 + 5 = 8 \]
\[ |0 - 8| = | -8 | = 8 \]
Since \( 8 \leq 8 \), this interval satisfies the inequality.

**Interval 3: \( [\frac{3}{2}, 8) \)**
- Choose \( x = 2 \):
\[ |2(2) - 3| + |2 + 5| = | 4 - 3 | + | 7 | = 1 + 7 = 8 \]
\[ |2 - 8| = | -6 | = 6 \]
Since \( 8 \nleq 6 \), this interval does not satisfy the inequality.

**Interval 4: \( [8, \infty) \)**
- Choose \( x = 9 \):
\[ |2(9) - 3| + |9 + 5| = | 18 - 3 | + | 14 | = 15 + 14 = 29 \]
\[ |9 - 8| = | 1 | = 1 \]
Since \( 29 \nleq 1 \), this interval does not satisfy the inequality.

### Step 4: Determine the solution set
The only interval that satisfies the inequality is \( [-5, \frac{3}{2}] \).

### Step 5: Count the integers in the solution set
The integers in the interval \( [-5, \frac{3}{2}] \) are:
- \( -5, -4, -3, -2, -1, 0, 1 \)

Thus, the number of integers satisfying the inequality is **7**.

### Final Answer
The number of integers satisfying the inequality \( |2x-3| + |x+5| \leq |x-8| \) is **7**.

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