Question

on a line - segment, u is between t and v. if tu = 9, uv=x + 11, and tv = 6x - 19, what is uv? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use segment - addition postulate

Since \(U\) is between \(T\) and \(V\), by the segment - addition postulate, \(TV=TU + UV\). We know that \(TU = 9\), \(UV=x + 11\), and \(TV=6x-19\). So, \(6x-19=9+(x + 11)\).

Step2: Simplify the right - hand side of the equation

First, simplify the right - hand side: \(9+(x + 11)=x+20\). So the equation becomes \(6x-19=x + 20\).

Step3: Solve for \(x\)

Subtract \(x\) from both sides: \(6x-x-19=x - x+20\), which simplifies to \(5x-19 = 20\). Then add 19 to both sides: \(5x-19 + 19=20 + 19\), so \(5x=39\), and \(x=\frac{39}{5}\).

Step4: Find \(UV\)

Substitute \(x=\frac{39}{5}\) into the expression for \(UV\). \(UV=x + 11=\frac{39}{5}+11\). Rewrite 11 as \(\frac{55}{5}\), then \(UV=\frac{39}{5}+\frac{55}{5}=\frac{39 + 55}{5}=\frac{94}{5}=18\frac{4}{5}\).

Answer:

\(18\frac{4}{5}\)