In the inequation ` 2+ ( 3x-1)/( 5) le (2x- 1)/( 4) + 3, ` write the greatest value of x, when
x is a natural number

Answer

To solve the inequation \( 2 + \frac{3x - 1}{5} \leq \frac{2x - 1}{4} + 3 \) and find the greatest value of \( x \) when \( x \) is a natural number, we will follow these steps:

### Step 1: Simplify the Inequation
Start with the given inequation:
\[ 2 + \frac{3x - 1}{5} \leq \frac{2x - 1}{4} + 3 \]

### Step 2: Move Constants to One Side
Subtract \( 2 \) from both sides:
\[ \frac{3x - 1}{5} \leq \frac{2x - 1}{4} + 1 \]

### Step 3: Combine the Right Side
Convert \( 1 \) to a fraction with a common denominator:
\[ 1 = \frac{4}{4} \]
Thus, we have:
\[ \frac{3x - 1}{5} \leq \frac{2x - 1 + 4}{4} \]
This simplifies to:
\[ \frac{3x - 1}{5} \leq \frac{2x + 3}{4} \]

### Step 4: Eliminate the Fractions
To eliminate the fractions, multiply both sides by \( 20 \) (the least common multiple of 5 and 4):
\[ 20 \cdot \frac{3x - 1}{5} \leq 20 \cdot \frac{2x + 3}{4} \]
This simplifies to:
\[ 4(3x - 1) \leq 5(2x + 3) \]

### Step 5: Distribute
Distributing on both sides gives:
\[ 12x - 4 \leq 10x + 15 \]

### Step 6: Rearrange the Inequation
Now, move \( 10x \) to the left side and \( -4 \) to the right side:
\[ 12x - 10x \leq 15 + 4 \]
This simplifies to:
\[ 2x \leq 19 \]

### Step 7: Solve for \( x \)
Dividing both sides by 2 gives:
\[ x \leq \frac{19}{2} \]
Which simplifies to:
\[ x \leq 9.5 \]

### Step 8: Find the Greatest Natural Number
Since \( x \) must be a natural number, the greatest natural number less than or equal to \( 9.5 \) is:
\[ x = 9 \]

### Final Answer
The greatest value of \( x \) when \( x \) is a natural number is:
\[ \boxed{9} \]