Solve the inequation `3x + 8 gt 2 `If (i) x `in` N (ii) X `in` R and represent the solution on the number line

Explanation

To solve the inequation \(3x + 8 > 2\), we will follow these steps:

### Step 1: Rewrite the Inequation
Start with the given inequation:
\[ 3x + 8 > 2 \]

### Step 2: Isolate the Variable
Subtract 8 from both sides to isolate the term with \(x\):
\[ 3x > 2 - 8 \]
\[ 3x > -6 \]

### Step 3: Divide by the Coefficient of \(x\)
Now, divide both sides by 3 to solve for \(x\):
\[ x > \frac{-6}{3} \]
\[ x > -2 \]

### Step 4: Consider the Cases
Now we will consider the two cases given in the question:

#### Case (i): \(x \in \mathbb{N}\) (Natural Numbers)
Natural numbers are positive integers starting from 1. Since \(x > -2\), the natural numbers that satisfy this condition are:
\[ x = 1, 2, 3, 4, \ldots \]
So, the solution set in natural numbers is:
\[ \{1, 2, 3, 4, \ldots\} \]

#### Case (ii): \(x \in \mathbb{R}\) (Real Numbers)
For real numbers, since \(x > -2\), the solution set includes all real numbers greater than -2:
\[ x \in (-2, \infty) \]

### Step 5: Representing the Solution on the Number Line
1. **For \(x \in \mathbb{N}\)**:
- Mark points at 1, 2, 3, 4, etc. on the number line.
- The representation will show these points as discrete values.

2. **For \(x \in \mathbb{R}\)**:
- Draw a number line and indicate that the solution includes all numbers greater than -2.
- Use an open circle at -2 (indicating -2 is not included) and shade the line to the right towards positive infinity.

### Summary of Solutions
- **For \(x \in \mathbb{N}\)**: The solution is \(\{1, 2, 3, 4, \ldots\}\).
- **For \(x \in \mathbb{R}\)**: The solution is \(x > -2\) or \(x \in (-2, \infty)\).

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