If `x inZ^(-)`, find the solution set of each of the following inequations : `4(x+5)lt9`

Answer

To solve the inequality \( 4(x + 5) < 9 \) given that \( x \) belongs to the set of negative integers (\( \mathbb{Z}^(-) \)), we will follow these steps:

### Step 1: Write the inequality
We start with the inequality:
\[ 4(x + 5) < 9 \]

### Step 2: Distribute the 4
Distributing the 4 on the left side gives us:
\[ 4x + 20 < 9 \]

### Step 3: Isolate the variable term
Next, we isolate the term with \( x \) by subtracting 20 from both sides:
\[ 4x < 9 - 20 \]
This simplifies to:
\[ 4x < -11 \]

### Step 4: Divide by 4
Now, we divide both sides by 4 to solve for \( x \):
\[ x < -\frac{11}{4} \]

### Step 5: Convert to a mixed number
The fraction \( -\frac{11}{4} \) can be converted to a mixed number:
\[ -\frac{11}{4} = -2.75 \]

### Step 6: Identify the solution set
Since \( x \) must be a negative integer, we need to find all negative integers that are less than \( -2.75 \). The negative integers satisfying this condition are:
\[ -3, -4, -5, \ldots \]

Thus, the solution set in interval notation is:
\[ (-\infty, -3) \]

### Final Answer
The solution set is:
\[ (-\infty, -3) \]

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