If `x inN`, find the solution set of each of the following inequations : `-(1)/(2)gt(1)/(4)-(x)/(3)`

Answer

To solve the inequality \(-\frac{1}{2} > \frac{1}{4} - \frac{x}{3}\), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ -\frac{1}{2} > \frac{1}{4} - \frac{x}{3} \]

### Step 2: Move \(\frac{x}{3}\) to the left side
To isolate \(x\), we can add \(\frac{x}{3}\) to both sides:
\[ -\frac{1}{2} + \frac{x}{3} > \frac{1}{4} \]

### Step 3: Eliminate fractions by finding a common denominator
The common denominator for \(-\frac{1}{2}\) and \(\frac{1}{4}\) is 4. We rewrite the left side:
\[ -\frac{2}{4} + \frac{x}{3} > \frac{1}{4} \]

### Step 4: Move \(-\frac{2}{4}\) to the right side
Now we add \(\frac{2}{4}\) to both sides:
\[ \frac{x}{3} > \frac{1}{4} + \frac{2}{4} \]
This simplifies to:
\[ \frac{x}{3} > \frac{3}{4} \]

### Step 5: Multiply both sides by 3
To eliminate the fraction, we multiply both sides by 3:
\[ x > \frac{3}{4} \times 3 \]
This simplifies to:
\[ x > \frac{9}{4} \]

### Step 6: Convert \(\frac{9}{4}\) to a decimal
Calculating \(\frac{9}{4}\) gives:
\[ \frac{9}{4} = 2.25 \]

### Step 7: Determine the natural number solutions
Since \(x\) must be a natural number (i.e., a positive integer), the smallest natural number greater than \(2.25\) is \(3\). Therefore, the solution set is:
\[ x \in \{3, 4, 5, \ldots\} \]

### Final Answer
The solution set of the inequality is:
\[ x \in \mathbb{N} \text{ such that } x \geq 3 \]

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