Solve each of the following inequations and represent the solution set on the number line.
`6xle25, " where " (i)x inN," "(ii)x inZ`.

Answer

To solve the inequation \( 6x \leq 25 \) and represent the solution set on the number line, we will follow these steps:

### Step 1: Solve the Inequation
We start with the inequation:
\[ 6x \leq 25 \]
To isolate \( x \), we divide both sides by 6:
\[ x \leq \frac{25}{6} \]
Calculating \( \frac{25}{6} \):
\[ \frac{25}{6} \approx 4.1667 \]

### Step 2: Identify the Solution Set for \( x \in \mathbb{N} \)
Now, we need to find the natural numbers (denoted as \( \mathbb{N} \)) that satisfy \( x \leq 4.1667 \). The natural numbers are \( 1, 2, 3, 4, \ldots \).

The natural numbers less than or equal to \( 4.1667 \) are:
\[ 1, 2, 3, 4 \]

### Step 3: Represent the Solution Set on the Number Line
On the number line, we will mark the natural numbers \( 1, 2, 3, \) and \( 4 \) as follows:
- Draw a number line.
- Mark points for \( 1, 2, 3, 4 \).
- Use closed circles at \( 1, 2, 3, \) and \( 4 \) to indicate that these numbers are included in the solution set.

### Step 4: Identify the Solution Set for \( x \in \mathbb{Z} \)
Next, we find the integers (denoted as \( \mathbb{Z} \)) that satisfy \( x \leq 4.1667 \). The integers include both positive and negative whole numbers, as well as zero.

The integers less than or equal to \( 4.1667 \) are:
\[ \ldots, -3, -2, -1, 0, 1, 2, 3, 4 \]

### Step 5: Represent the Solution Set on the Number Line
On the number line, we will mark the integers \( \ldots, -3, -2, -1, 0, 1, 2, 3, 4 \):
- Draw a number line.
- Mark points for \( -3, -2, -1, 0, 1, 2, 3, 4 \).
- Use closed circles at \( -3, -2, -1, 0, 1, 2, 3, \) and \( 4 \) to indicate that these numbers are included in the solution set.

### Final Solution
1. For \( x \in \mathbb{N} \): The solution set is \( \{1, 2, 3, 4\} \).
2. For \( x \in \mathbb{Z} \): The solution set is \( \{\ldots, -3, -2, -1, 0, 1, 2, 3, 4\} \).