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Here are solutions for the questions found in your document:

### 1. Distance Between Two Points
The problem asks for the distance between points \( A(2, 3) \) and \( B(3, x) \) as a function of \( x \). Using the distance formula:

\[
d = \sqrt{(3 - 2)^2 + (x - 3)^2} = \sqrt{1 + (x - 3)^2}
\]

Simplifying this, none of the given choices seem to directly match. However, rewriting the expression \(\sqrt{1 + (x - 3)^2}\) could potentially match one of the options after further manipulation.

### 2. Equation of the Line
The line passes through the origin \((0, 0)\) and the point \((8, 4)\). The slope (\(m\)) is calculated as:

\[
m = \frac{4 - 0}{8 - 0} = \frac{1}{2}
\]

Thus, the equation in standard form is \( 2y - x = 0 \), matching choice (c).

### 3. Area of a Triangle with a Negative Integer Value of \(x\)
Given vertices \(A(4, 5)\), \(B(7, 4)\), and \(C(x, 2)\), the area of the triangle can be calculated using:

\[
\text{Area} = \frac{1}{2} |4(4 - 2) + 7(2 - 5) + x(5 - 4)| = 7
\]

Solving this equation, the correct negative integer value for \(x\) is \(-5\) (choice a).

### 4. Area of a Triangle with Positive Integer Value of \(x\)
Given vertices \(A(-4, 2)\), \(B(x, 3)\), and \(C(-2, -2)\), the formula for the area leads to finding \(x = 5\), matching choice (a).

### 5. Equation of the Line Through the Intersection and Parallel to Given Line
Finding the point of intersection and determining the parallel line equation that matches one of the options, the correct equation is \(y = -2x - 23\) (choice e).

Would you like any further clarifications, or need detailed steps for any specific question?

### Related Questions
1. How do you find the equation of a line given two points?
2. What is the formula for the area of a triangle using coordinates?
3. How do you determine if lines are parallel or perpendicular?
4. What is the process for calculating the distance between two points in a coordinate plane?
5. How can you determine if three points are collinear?

**Tip:** Always double-check your formulas and calculations when solving coordinate geometry problems to avoid errors!

Given the distance between two points A(2, 3) and B(3, x), what is the expression for the distance as a function of x?

Learn how to calculate the distance between points A(2, 3) and B(3, x) using the distance formula. This problem covers essential concepts in coordinate geometry, suitable for high school students.

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