Question

point n is on line segment \\( \overline{mo} \\). given \\( mn = x \\), \\( mo = 3x - 6 \\), and \\( no = x + 1 \\), determine the numerical length of \\( \overline{mo} \\).

Explanation:

Step1: Use the segment - addition postulate

Since \(N\) is on \(\overline{MO}\), we have \(MN + NO=MO\).
Substitute \(MN = x\), \(MO = 3x - 6\), and \(NO = x + 1\) into the equation: \(x+(x + 1)=3x - 6\).

Step2: Simplify the left - hand side of the equation

Combine like terms on the left - hand side: \(2x+1 = 3x - 6\).

Step3: Solve for \(x\)

Subtract \(2x\) from both sides: \(1=x - 6\).
Then add \(6\) to both sides: \(x=7\).

Step4: Find the length of \(MO\)

Substitute \(x = 7\) into the expression for \(MO\): \(MO=3x - 6\).
\(MO=3\times7-6\).
First, calculate \(3\times7 = 21\), then \(21-6=15\).

Answer:

\(15\)