Introduction:
To find the point on the x-axis that is at a distance of 10 units from point A(11, -8), we need to use the distance formula and apply it to the given problem. The distance formula is derived from the Pythagorean theorem and can be used to find the distance between two points in a coordinate plane.

Step 1: Understand the problem:
The given problem asks us to find a point on the x-axis that is 10 units away from point A(11, -8). This means that the y-coordinate of the point on the x-axis will be 0.

Step 2: Recall the distance formula:
The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Step 3: Apply the distance formula:
In this problem, we have point A(11, -8) and we need to find a point on the x-axis that is 10 units away from A. Let's assume the x-coordinate of the point on the x-axis is x.

Using the distance formula, we can write the equation:
10 = sqrt((x - 11)^2 + (0 - (-8))^2)

Simplifying this equation, we get:
100 = (x - 11)^2 + 64

Step 4: Solve the equation:
Expanding the equation, we get:
100 = x^2 - 22x + 121 + 64

Combining like terms, we get:
x^2 - 22x + 185 = 0

Step 5: Find the solutions:
To find the solutions of this quadratic equation, we can either factor it or use the quadratic formula. In this case, the equation cannot be factored easily, so we will use the quadratic formula.

The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our equation, the coefficients are:
a = 1, b = -22, c = 185

Substituting these values into the quadratic formula, we get:
x = (-(-22) ± sqrt((-22)^2 - 4(1)(185))) / (2(1))

Simplifying further, we get:
x = (22 ± sqrt(484 - 740)) / 2

Since the discriminant (b^2 - 4ac) is negative, the quadratic equation has no real solutions. This means that there are no points on the x-axis that are 10 units away from point A(11, -8).

Conclusion:
In conclusion, there are no points on the x-axis that are at a distance of 10 units from the point A(11, -8). This is because the quadratic equation derived from the distance formula has no real solutions.