Determine the value of 'x' for which the area of the triangle formed by joining the points `(x,4)`, `(0,8)` and `(-1,3)` will be `4(1)/(2)` square units.

Answer

To determine the value of 'x' for which the area of the triangle formed by the points \((x, 4)\), \((0, 8)\), and \((-1, 3)\) is \(4.5\) square units, we will use the formula for the area of a triangle given by its vertices.

### Step 1: Write the formula for the area of a triangle
The area \(A\) of a triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 2: Substitute the coordinates into the formula
Here, the coordinates are:
- \(P(x, 4)\) where \(x_1 = x\) and \(y_1 = 4\)
- \(Q(0, 8)\) where \(x_2 = 0\) and \(y_2 = 8\)
- \(R(-1, 3)\) where \(x_3 = -1\) and \(y_3 = 3\)

Substituting these values into the area formula:
\[ A = \frac{1}{2} \left| x(8 - 3) + 0(3 - 4) + (-1)(4 - 8) \right| \]

### Step 3: Simplify the expression
Calculating the terms:
\[ A = \frac{1}{2} \left| x(5) + 0 + (-1)(-4) \right| \]
\[ A = \frac{1}{2} \left| 5x + 4 \right| \]

### Step 4: Set the area equal to \(4.5\)
We know the area is \(4.5\) square units, so we set up the equation:
\[ \frac{1}{2} \left| 5x + 4 \right| = 4.5 \]

### Step 5: Solve for \(|5x + 4|\)
Multiplying both sides by \(2\):
\[ \left| 5x + 4 \right| = 9 \]

### Step 6: Solve the absolute value equation
This gives us two cases to consider:
1. \(5x + 4 = 9\)
2. \(5x + 4 = -9\)

#### Case 1: \(5x + 4 = 9\)
\[ 5x = 9 - 4 \]
\[ 5x = 5 \]
\[ x = 1 \]

#### Case 2: \(5x + 4 = -9\)
\[ 5x = -9 - 4 \]
\[ 5x = -13 \]
\[ x = -\frac{13}{5} \]

### Conclusion
The values of \(x\) for which the area of the triangle is \(4.5\) square units are:
\[ x = 1 \quad \text{and} \quad x = -\frac{13}{5} \]