The given compound inequality is:
[tex]-6x+7\lt55\text{ and }5x-4\leq16[/tex]Solve the first inequality:
[tex]\begin{gathered} -6x+7<55 \\ -6x<55-7 \\ -6x<48 \\ \text{ Divide both sides by }-6\text{ and reverse the inequality sign:} \\ -\frac{6x}{-6}\gt\frac{48}{-6} \\ x\gt-8 \end{gathered}[/tex]Solve the second inequality:
[tex]\begin{gathered} 5x-4\leq16 \\ \text{ Add }4\text{ to both sides of the inequality:} \\ 5x-4+4\leq16+4 \\ 5x\leq20 \\ \frac{5x}{5}\leq\frac{20}{5} \\ x\leq4 \end{gathered}[/tex]Therefore, the correct answer is choice C:
-8 < x ≤ 4
