Solve the following inequation and write down the solution set:
`11x-4 lt 15x + 4 le 13x+14, x in W` .

Answer

To solve the given inequation \( 11x - 4 < 15x + 4 \leq 13x + 14 \) where \( x \in W \) (the set of whole numbers), we will break it down into two parts.

### Step 1: Rewrite the Inequation
The given inequation is:
\[ 11x - 4 < 15x + 4 \leq 13x + 14 \]

### Step 2: Solve the Left Inequality
First, we will solve the left part of the inequality:
\[ 11x - 4 < 15x + 4 \]
Subtract \( 11x \) from both sides:
\[ -4 < 4x + 4 \]
Next, subtract \( 4 \) from both sides:
\[ -4 - 4 < 4x \]
This simplifies to:
\[ -8 < 4x \]
Now, divide both sides by \( 4 \):
\[ -2 < x \]
This can be rewritten as:
\[ x > -2 \quad \text{(Equation 1)} \]

### Step 3: Solve the Right Inequality
Now, we will solve the right part of the inequality:
\[ 15x + 4 \leq 13x + 14 \]
Subtract \( 13x \) from both sides:
\[ 2x + 4 \leq 14 \]
Next, subtract \( 4 \) from both sides:
\[ 2x \leq 10 \]
Now, divide both sides by \( 2 \):
\[ x \leq 5 \quad \text{(Equation 2)} \]

### Step 4: Combine the Results
Now we have two inequalities:
1. \( x > -2 \)
2. \( x \leq 5 \)

### Step 5: Write the Solution Set
Combining these inequalities, we find:
\[ -2 < x \leq 5 \]
Since \( x \) must be a whole number (as \( x \in W \)), the solution set in whole numbers is:
\[ \{0, 1, 2, 3, 4, 5\} \]

### Final Answer
Thus, the solution set is:
\[ \{0, 1, 2, 3, 4, 5\} \]