If ` xinZ^(+)`, find the solution set of each of the following inequations : `5gt(2)/(3)x`

Answer

To solve the inequality \( 5 > \frac{2}{3}x \) where \( x \) belongs to the set of positive integers \( \mathbb{Z}^+ \), we will follow these steps:

### Step 1: Write the inequality
We start with the given inequality:
\[ 5 > \frac{2}{3}x \]

### Step 2: Eliminate the fraction
To eliminate the fraction, we can multiply both sides of the inequality by 3:
\[ 3 \times 5 > 2x \]
This simplifies to:
\[ 15 > 2x \]

### Step 3: Rearrange the inequality
Now, we can rearrange the inequality to isolate \( x \):
\[ 2x < 15 \]

### Step 4: Divide by 2
Next, we divide both sides by 2 to solve for \( x \):
\[ x < \frac{15}{2} \]
Calculating \( \frac{15}{2} \) gives us:
\[ x < 7.5 \]

### Step 5: Identify the solution set
Since \( x \) must be a positive integer, we need to find all positive integers less than 7.5. The positive integers are:
\[ 1, 2, 3, 4, 5, 6, 7 \]

### Final Solution Set
Thus, the solution set for the inequality \( 5 > \frac{2}{3}x \) where \( x \) belongs to the set of positive integers is:
\[ \{ 1, 2, 3, 4, 5, 6, 7 \} \]

---