Solve `(3)/(7)-(x)/(3)gt(-1)/(4),x inN`

Answer

To solve the inequality \(\frac{3}{7} - \frac{x}{3} > -\frac{1}{4}\) where \(x\) is a natural number, we can follow these steps:

### Step 1: Rewrite the Inequality
Start with the original inequality:
\[ \frac{3}{7} - \frac{x}{3} > -\frac{1}{4} \]

### Step 2: Find a Common Denominator
The least common multiple (LCM) of the denominators 7, 3, and 4 is 84. We will rewrite each term with a denominator of 84.

- For \(\frac{3}{7}\):
\[ \frac{3}{7} = \frac{3 \times 12}{7 \times 12} = \frac{36}{84} \]

- For \(-\frac{x}{3}\):
\[ -\frac{x}{3} = -\frac{x \times 28}{3 \times 28} = -\frac{28x}{84} \]

- For \(-\frac{1}{4}\):
\[ -\frac{1}{4} = -\frac{1 \times 21}{4 \times 21} = -\frac{21}{84} \]

Now, we can rewrite the inequality as:
\[ \frac{36}{84} - \frac{28x}{84} > -\frac{21}{84} \]

### Step 3: Combine the Left Side
Combine the fractions on the left side:
\[ \frac{36 - 28x}{84} > -\frac{21}{84} \]

### Step 4: Eliminate the Denominator
Multiply both sides by 84 (since 84 is positive, the direction of the inequality remains the same):
\[ 36 - 28x > -21 \]

### Step 5: Rearrange the Inequality
Now, isolate the term with \(x\):
\[ 36 + 21 > 28x \]
\[ 57 > 28x \]

### Step 6: Solve for \(x\)
Divide both sides by 28:
\[ \frac{57}{28} > x \]

### Step 7: Approximate the Value
Calculating \(\frac{57}{28}\):
\[ \frac{57}{28} \approx 2.0357 \]

This means:
\[ x < 2.0357 \]

### Step 8: Determine Natural Numbers
Since \(x\) must be a natural number (positive integers), the possible values for \(x\) are:
\[ x = 1, 2 \]

### Final Solution
Thus, the solution set for \(x\) is:
\[ \{1, 2\} \]

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