Solve and graph the solution set on a number line :
`x-5 lt -2, x in N`

Answer

To solve the inequality \( x - 5 < -2 \) where \( x \) belongs to the natural numbers (\( \mathbb{N} \)), we will follow these steps:

### Step 1: Solve the Inequality
Start with the inequality:
\[ x - 5 < -2 \]

Add 5 to both sides to isolate \( x \):
\[ x < -2 + 5 \]

This simplifies to:
\[ x < 3 \]

### Step 2: Identify Natural Numbers
Since \( x \) must be a natural number, we need to find the natural numbers that are less than 3. The natural numbers start from 1 and go upwards (1, 2, 3, ...).

The natural numbers less than 3 are:
- 1
- 2

### Step 3: Write the Solution Set
Thus, the solution set for the inequality \( x - 5 < -2 \) where \( x \in \mathbb{N} \) is:
\[ \{1, 2\} \]

### Step 4: Graph the Solution on a Number Line
To graph the solution on a number line:
1. Draw a horizontal line and mark points for the relevant numbers.
2. Indicate the numbers 1 and 2 on the line.
3. Since both 1 and 2 are included in the solution, we will use closed circles at these points.

The number line will look like this:

```
<---|---|---|---|---|---|---|---|---|--->
-1 0 1 2 3 4 5
● ●
```

### Final Solution
The solution set is \( \{1, 2\} \) and it is represented on the number line with closed circles at 1 and 2.

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