Solve `3-(x)/(4)lt2x-9lt12-(x)/(2),x inZ`

Answer

To solve the inequality \( 3 - \frac{x}{4} < 2x - 9 < 12 - \frac{x}{2} \) where \( x \in \mathbb{Z} \), we will break it down into two parts.

### Step 1: Solve the first inequality
We start with the first part of the compound inequality:

\[ 3 - \frac{x}{4} < 2x - 9 \]

**Hint:** Rearranging the inequality will help isolate \( x \).

1. Multiply through by 4 to eliminate the fraction:
\[ 4(3) - x < 4(2x - 9) \]
This simplifies to:
\[ 12 - x < 8x - 36 \]

2. Rearranging gives:
\[ 12 + 36 < 8x + x \]
\[ 48 < 9x \]

3. Divide by 9:
\[ \frac{48}{9} < x \]
Simplifying \( \frac{48}{9} \) gives:
\[ x > \frac{16}{3} \approx 5.33 \]

### Step 2: Solve the second inequality
Now we solve the second part of the compound inequality:

\[ 2x - 9 < 12 - \frac{x}{2} \]

**Hint:** Again, rearranging will help isolate \( x \).

1. Multiply through by 2 to eliminate the fraction:
\[ 2(2x - 9) < 2(12 - \frac{x}{2}) \]
This simplifies to:
\[ 4x - 18 < 24 - x \]

2. Rearranging gives:
\[ 4x + x < 24 + 18 \]
\[ 5x < 42 \]

3. Divide by 5:
\[ x < \frac{42}{5} \approx 8.4 \]

### Step 3: Combine the results
From the two inequalities we have:

1. \( x > \frac{16}{3} \) (or approximately 5.33)
2. \( x < \frac{42}{5} \) (or approximately 8.4)

Since \( x \) must be an integer, we find the integer solutions between these two bounds:

- The integers greater than \( 5.33 \) are \( 6, 7, 8 \).
- The integers less than \( 8.4 \) are \( 6, 7, 8 \).

Thus, the solution set is:

\[ x \in \{6, 7, 8\} \]

### Final Answer
The final solution set is \( x = 6, 7, 8 \).

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