If the replacement set is the of natural numbers (N), find the solution set of :
`8 - x le 4x - 2`.

Answer

To solve the inequality \( 8 - x \leq 4x - 2 \) with the replacement set being the natural numbers \( N \), we will follow these steps:

### Step 1: Rewrite the Inequality
Start with the original inequality:
\[ 8 - x \leq 4x - 2 \]

### Step 2: Add \( x \) to Both Sides
To eliminate \( -x \) from the left side, add \( x \) to both sides:
\[ 8 \leq 4x - 2 + x \]
This simplifies to:
\[ 8 \leq 5x - 2 \]

### Step 3: Add 2 to Both Sides
Next, add 2 to both sides to isolate the term with \( x \):
\[ 8 + 2 \leq 5x \]
This simplifies to:
\[ 10 \leq 5x \]

### Step 4: Divide Both Sides by 5
Now, divide both sides by 5 to solve for \( x \):
\[ \frac{10}{5} \leq x \]
This simplifies to:
\[ 2 \leq x \]
or
\[ x \geq 2 \]

### Step 5: Determine the Solution Set
Since the replacement set is the natural numbers \( N \) (which includes 1, 2, 3, 4, ...), we need to find the natural numbers that satisfy \( x \geq 2 \). The natural numbers that satisfy this condition are:
\[ \{2, 3, 4, 5, \ldots\} \]

### Final Solution
The solution set is:
\[ \{x \in N : x \geq 2\} = \{2, 3, 4, 5, \ldots\} \]

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