If `x inN`, find the solution set of each of the following inequations : `(4)/(7)-(x)/(4)gt-2`

Answer

To solve the inequality \( \frac{4}{7} - \frac{x}{4} > -2 \) for \( x \in \mathbb{N} \), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ \frac{4}{7} - \frac{x}{4} > -2 \]

### Step 2: Eliminate the fractions
To eliminate the fractions, we can find a common denominator. The common denominator for 7 and 4 is 28. Multiply the entire inequality by 28:
\[ 28 \left( \frac{4}{7} \right) - 28 \left( \frac{x}{4} \right) > 28 \times (-2) \]

### Step 3: Simplify the expression
Calculating each term, we get:
\[ 16 - 7x > -56 \]

### Step 4: Rearrange the inequality
Now, we will move -56 to the left side:
\[ 16 + 56 > 7x \]
This simplifies to:
\[ 72 > 7x \]

### Step 5: Isolate \( x \)
Now, divide both sides by 7 to isolate \( x \):
\[ \frac{72}{7} > x \]
This can be rewritten as:
\[ x < \frac{72}{7} \]

### Step 6: Calculate \( \frac{72}{7} \)
Calculating \( \frac{72}{7} \):
\[ \frac{72}{7} \approx 10.2857 \]

### Step 7: Identify natural numbers
Since \( x \) must be a natural number, we take all natural numbers less than \( 10.2857 \). The natural numbers less than 10.2857 are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]

### Final Solution
Thus, the solution set is:
\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

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