Solution set for inequality 5x-3<3x+1,`xinN`is-

Options

A. `(-infty,2)`
B. {0,1,2}
C. {1}
D. `phi`

Answer

To solve the inequality \(5x - 3 < 3x + 1\) where \(x \in \mathbb{N}\) (natural numbers), we will follow these steps:

### Step 1: Rearranging the Inequality
Start with the given inequality:
\[ 5x - 3 < 3x + 1 \]

### Step 2: Move all terms involving \(x\) to one side
Subtract \(3x\) from both sides:
\[ 5x - 3 - 3x < 1 \]
This simplifies to:
\[ 2x - 3 < 1 \]

### Step 3: Move constant terms to the other side
Add \(3\) to both sides:
\[ 2x < 1 + 3 \]
This simplifies to:
\[ 2x < 4 \]

### Step 4: Solve for \(x\)
Divide both sides by \(2\):
\[ x < \frac{4}{2} \]
This simplifies to:
\[ x < 2 \]

### Step 5: Consider the domain of \(x\)
Since \(x\) must be a natural number (\(x \in \mathbb{N}\)), the possible values for \(x\) are \(1, 2, 3, \ldots\). However, from the inequality \(x < 2\), the only natural number that satisfies this condition is:
\[ x = 1 \]

### Conclusion
Thus, the solution set for the inequality \(5x - 3 < 3x + 1\) where \(x \in \mathbb{N}\) is:
\[ \{1\} \]