If ` xinZ^(+)`, find the solution set of each of the following inequations : `8-xgt(1)/(3)`

Answer

To solve the inequality \( 8 - x > \frac{1}{3} \) where \( x \) belongs to the set of positive integers \( \mathbb{Z}^+ \), we can follow these steps:

### Step 1: Write the inequality
We start with the given inequality:
\[ 8 - x > \frac{1}{3} \]

### Step 2: Isolate \( x \)
To isolate \( x \), we can first subtract 8 from both sides:
\[ -x > \frac{1}{3} - 8 \]
Calculating the right side:
\[ \frac{1}{3} - 8 = \frac{1}{3} - \frac{24}{3} = \frac{1 - 24}{3} = \frac{-23}{3} \]
So we have:
\[ -x > \frac{-23}{3} \]

### Step 3: Multiply by -1
When we multiply or divide an inequality by a negative number, we must reverse the inequality sign:
\[ x < \frac{23}{3} \]

### Step 4: Convert to decimal
Calculating \( \frac{23}{3} \):
\[ \frac{23}{3} \approx 7.67 \]

### Step 5: Identify the solution set
Since \( x \) must be a positive integer, we consider all positive integers less than \( 7.67 \). The positive integers satisfying this condition are:
\[ 1, 2, 3, 4, 5, 6, 7 \]

### Final Solution Set
Thus, the solution set for the inequality \( 8 - x > \frac{1}{3} \) where \( x \in \mathbb{Z}^+ \) is:
\[ \{ 1, 2, 3, 4, 5, 6, 7 \} \]

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