Question

q is the midpoint of \\(\overline{pr}\\). if \\(pq = x + 2\\) and \\(pr = 3x - 2\\), what is \\(pq\\)? image of a line segment with points p, q, r; pq is labeled \\(x + 2\\), pr is labeled \\(3x - 2\\) simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since Q is the midpoint of \( \overline{PR} \), \( PR = 2 \times PQ \). So we have the equation \( 3x - 2 = 2(x + 2) \).

Step2: Solve for x

Expand the right side: \( 3x - 2 = 2x + 4 \). Subtract \( 2x \) from both sides: \( 3x - 2x - 2 = 2x - 2x + 4 \), which simplifies to \( x - 2 = 4 \). Then add 2 to both sides: \( x - 2 + 2 = 4 + 2 \), so \( x = 6 \).

Step3: Find PQ

Substitute \( x = 6 \) into \( PQ = x + 2 \). So \( PQ = 6 + 2 = 8 \).

Answer:

8