If `x in W,` find the solution set of `(3)/(5) x - ( 2x -1)/(3) gt1.`

Answer

To solve the inequality \(\frac{3}{5}x - \frac{2x - 1}{3} > 1\) where \(x\) belongs to the set of whole numbers \(W\), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ \frac{3}{5}x - \frac{2x - 1}{3} > 1 \]
We can rearrange this by moving \(1\) to the left side:
\[ \frac{3}{5}x - \frac{2x - 1}{3} - 1 > 0 \]

### Step 2: Find a common denominator
The common denominator for \(5\) and \(3\) is \(15\). We will rewrite each term with this common denominator:
\[ \frac{3}{5}x = \frac{9}{15}x, \quad \frac{2x - 1}{3} = \frac{10x - 5}{15} \]
Thus, the inequality becomes:
\[ \frac{9x}{15} - \frac{10x - 5}{15} - 1 > 0 \]

### Step 3: Combine the fractions
Now, we can combine the fractions:
\[ \frac{9x - (10x - 5)}{15} - 1 > 0 \]
This simplifies to:
\[ \frac{9x - 10x + 5}{15} - 1 > 0 \]
\[ \frac{-x + 5}{15} - 1 > 0 \]

### Step 4: Move \(1\) to the other side
Next, we can move \(1\) to the right side:
\[ \frac{-x + 5}{15} > 1 \]

### Step 5: Multiply through by \(15\)
Now we multiply both sides by \(15\) (since \(15 > 0\), the inequality sign remains the same):
\[ -x + 5 > 15 \]

### Step 6: Solve for \(x\)
Rearranging gives us:
\[ -x > 10 \quad \Rightarrow \quad x < -10 \]

### Step 7: Consider the domain of \(x\)
Since \(x\) must be a whole number (i.e., \(x \in W\)), and the solution \(x < -10\) does not include any whole numbers (as whole numbers start from \(0\)), we conclude that there are no solutions.

### Final Answer
Thus, the solution set is:
\[ \text{No solution} \]