Solve the inequation `3x - 2 lt 5 ` If (i) x`in` N (ii) x `in` R and represent the solution on the number line

Explanation

To solve the inequation \(3x - 2 < 5\), we will follow these steps:

### Step 1: Solve the inequation
1. Start with the original inequation:
\[ 3x - 2 < 5 \]

2. Add 2 to both sides to isolate the term with \(x\):
\[ 3x - 2 + 2 < 5 + 2 \]
This simplifies to:
\[ 3x < 7 \]

3. Now, divide both sides by 3 to solve for \(x\):
\[ x < \frac{7}{3} \]
This can be approximated as:
\[ x < 2.33 \]

### Step 2: Consider the two cases for \(x\)

#### Case (i): \(x \in \mathbb{N}\) (Natural Numbers)
- Natural numbers are positive integers: \(1, 2, 3, \ldots\)
- Since \(x < 2.33\), the natural numbers that satisfy this condition are:
\[ x = 1, 2 \]
- Therefore, the solution in this case is:
\[ x \in \{1, 2\} \]

#### Case (ii): \(x \in \mathbb{R}\) (Real Numbers)
- For real numbers, \(x\) can take any value less than \(2.33\).
- Therefore, the solution in this case is:
\[ x < \frac{7}{3} \quad \text{or} \quad x \in (-\infty, \frac{7}{3}) \]

### Step 3: Represent the solutions on the number line

- For \(x \in \mathbb{N}\):
- Mark the points \(1\) and \(2\) on the number line. Use closed dots to indicate that these values are included in the solution.

- For \(x \in \mathbb{R}\):
- Draw a number line and shade the region to the left of \(\frac{7}{3}\) (approximately \(2.33\)). Use an open circle at \(\frac{7}{3}\) to indicate that this value is not included.

### Summary of the Solutions
- (i) For \(x \in \mathbb{N}\): \(x \in \{1, 2\}\)
- (ii) For \(x \in \mathbb{R}\): \(x \in (-\infty, \frac{7}{3})\)