If `x inZ^(-)`, find the solution set of each of the following inequations : `3xgt-12`

Answer

To solve the inequation \(3x > -12\) given that \(x\) belongs to the set of negative integers (\(Z^-\)), we can follow these steps:

### Step 1: Rewrite the Inequation
We start with the inequation:
\[ 3x > -12 \]

### Step 2: Divide Both Sides by 3
To isolate \(x\), we divide both sides of the inequation by 3:
\[ x > \frac{-12}{3} \]
This simplifies to:
\[ x > -4 \]

### Step 3: Identify the Set of Negative Integers
Since \(x\) must be a negative integer, we need to consider the negative integers that are greater than \(-4\). The negative integers are:
\[ -1, -2, -3, -4, -5, \ldots \]

### Step 4: Find the Valid Negative Integers
From the inequality \(x > -4\), the negative integers that satisfy this condition are:
\[ -3, -2, -1 \]
(Note that \(-4\) is not included because the inequality is strict.)

### Step 5: Write the Solution Set
Thus, the solution set of the inequation \(3x > -12\) where \(x\) belongs to negative integers is:
\[ \{-3, -2, -1\} \]