Answer
$x\ge81
$
Work Step by Step
Let $x$ be the final exam score.
The conditions of the problem translate to
\begin{array}{l}\require{cancel}
\dfrac{72+67+82+79+2x}{6}\ge77
.\end{array}
Using the properties of inequality, then
\begin{array}{l}\require{cancel}
6\cdot\dfrac{72+67+82+79+2x}{6}\ge6\cdot77
\\\\
72+67+82+79+2x\ge462
\\\\
300+2x\ge462
\\\\
2x\ge462-300
\\\\
2x\ge162
\\\\
x\ge\dfrac{162}{2}
\\\\
x\ge81
.\end{array}
