Find the solution set of each of the following inequations : `10lt4x-5lt21,x inN`

Answer

To solve the inequality \(10 < 4x - 5 < 21\) where \(x\) belongs to natural numbers (\(N\)), we can break it down into two parts.

### Step 1: Solve the left part of the inequality
We start with the left part of the inequality:
\[ 10 < 4x - 5 \]
To isolate \(4x\), we add \(5\) to both sides:
\[ 10 + 5 < 4x \]
\[ 15 < 4x \]
Now, we divide both sides by \(4\):
\[ \frac{15}{4} < x \]
Calculating \(\frac{15}{4}\):
\[ \frac{15}{4} = 3.75 \]
Thus, we have:
\[ x > 3.75 \]

### Step 2: Solve the right part of the inequality
Now, we solve the right part of the inequality:
\[ 4x - 5 < 21 \]
Again, we add \(5\) to both sides:
\[ 4x < 21 + 5 \]
\[ 4x < 26 \]
Now, we divide both sides by \(4\):
\[ x < \frac{26}{4} \]
Calculating \(\frac{26}{4}\):
\[ \frac{26}{4} = 6.5 \]
Thus, we have:
\[ x < 6.5 \]

### Step 3: Combine the results
Now we combine the two results:
\[ 3.75 < x < 6.5 \]
Since \(x\) must be a natural number, the possible values for \(x\) that satisfy this inequality are \(4, 5, 6\).

### Final Answer
The solution set is:
\[ x \in \{4, 5, 6\} \]