Solve the following inequation and represent the solution set on a number line.
`-8(1)/(2)lt-1/(2)-4xle7 1/(2),x inI`

Answer

To solve the given inequation \(-8 \frac{1}{2} < -\frac{1}{2} - 4x \leq 7 \frac{1}{2}\), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
First, we convert the mixed numbers into improper fractions for easier calculations:
- \(-8 \frac{1}{2} = -\frac{17}{2}\)
- \(7 \frac{1}{2} = \frac{15}{2}\)

So, the inequation becomes:
\[ -\frac{17}{2} < -\frac{1}{2} - 4x \leq \frac{15}{2} \]

### Step 2: Add \(\frac{1}{2}\) to all parts of the inequality
Next, we add \(\frac{1}{2}\) to all sides of the inequality:
\[ -\frac{17}{2} + \frac{1}{2} < -4x \leq \frac{15}{2} + \frac{1}{2} \]
Calculating the left side:
\[ -\frac{17}{2} + \frac{1}{2} = -\frac{16}{2} = -8 \]
Calculating the right side:
\[ \frac{15}{2} + \frac{1}{2} = \frac{16}{2} = 8 \]
Thus, we have:
\[ -8 < -4x \leq 8 \]

### Step 3: Divide the entire inequality by -4
Now, we divide the entire inequality by -4. Remember that dividing by a negative number reverses the inequality signs:
\[ \frac{-8}{-4} > x \geq \frac{8}{-4} \]
Calculating the values:
\[ 2 > x \geq -2 \]
This can be rewritten as:
\[ -2 \leq x < 2 \]

### Step 4: Identify the integer solutions
The integer solutions for \(x\) that satisfy \(-2 \leq x < 2\) are:
\[ x = -2, -1, 0, 1 \]

### Step 5: Represent the solution set on a number line
To represent the solution set on a number line, we will mark the points -2, -1, 0, and 1. The point -2 will be included (closed circle) and the point 2 will not be included (open circle).

### Final Answer
The solution set is:
\[ x = -2, -1, 0, 1 \]