Solve `12+3(4x-3)gtx-23`, where `x inZ^(-)`

Answer

To solve the inequality \( 12 + 3(4x - 3) > x - 23 \) where \( x \) belongs to negative integers (\( \mathbb{Z}^(-) \)), follow these steps:

### Step 1: Expand the left side of the inequality
Start by distributing the \( 3 \) in the expression \( 3(4x - 3) \).

\[ 12 + 3(4x - 3) = 12 + 12x - 9 \]

### Step 2: Simplify the expression
Combine the constant terms on the left side.

\[ 12 - 9 + 12x > x - 23 \]
\[ 3 + 12x > x - 23 \]

### Step 3: Move all \( x \) terms to one side
Subtract \( x \) from both sides to isolate the \( x \) terms.

\[ 3 + 12x - x > -23 \]
\[ 3 + 11x > -23 \]

### Step 4: Move the constant to the other side
Subtract \( 3 \) from both sides.

\[ 11x > -23 - 3 \]
\[ 11x > -26 \]

### Step 5: Solve for \( x \)
Divide both sides by \( 11 \).

\[ x > -\frac{26}{11} \]
\[ x > -2.36 \]

### Step 6: Identify the negative integer solutions
Since \( x \) must be a negative integer, the possible values greater than \( -2.36 \) in negative integers are \( -2 \) and \( -1 \).

### Final Solution
The solution set is:
\[ \{-2, -1\} \]

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