Answer
$82$
Work Step by Step
Let $x$ be the final exam score.
The conditions of the problem translate to
\begin{array}{l}\require{cancel}
\dfrac{78+65+82+79+x+x}{6}\ge78
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{78+65+82+79+2x}{6}\ge78
\\
6\cdot\dfrac{78+65+82+79+2x}{6}\ge78\cdot6
\\
78+65+82+79+2x\ge468
\\
304+2x\ge468
\\
2x\ge468-304
\\
2x\ge164
\\
x\ge\dfrac{164}{2}
\\
x\ge82
.\end{array}
Hence her minimum score in the final exam should be $
82
.$
