**Finding the value of x to minimize PR + RQ:**

To find the value of x that minimizes the sum of PR and RQ, we need to minimize the distance between P and R and the distance between R and Q. The distance between two points in a Cartesian plane can be calculated using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we need to minimize the distance PR + RQ, which can be represented as:

PR + RQ = √((x - 2)^2 + (0 - 3)^2) + √((x - 4)^2 + (0 - (-2))^2)

Simplifying this expression, we get:

PR + RQ = √((x - 2)^2 + 9) + √((x - 4)^2 + 4)

To minimize this expression, we can take the derivative with respect to x and set it equal to zero:

d/dx (PR + RQ) = (x - 2) / √((x - 2)^2 + 9) + (x - 4) / √((x - 4)^2 + 4) = 0

Solving this equation will give us the value of x that minimizes PR + RQ.

**Finding the value of x to maximize |PR - RQ|:**

To find the value of x that maximizes the absolute difference between PR and RQ, we need to maximize the difference between the distances PR and RQ. This can be represented as:

|PR - RQ| = |√((x - 2)^2 + 9) - √((x - 4)^2 + 4)|

To maximize this expression, we can take the derivative with respect to x and set it equal to zero:

d/dx |PR - RQ| = (x - 2) / √((x - 2)^2 + 9) - (x - 4) / √((x - 4)^2 + 4) = 0

Solving this equation will give us the value of x that maximizes |PR - RQ|.

Since the calculations for both cases involve derivatives and solving equations, the exact values of x to minimize PR + RQ and maximize |PR - RQ| can only be determined through numerical methods or approximation techniques.

Is it 4 for both ??