Consider the given inequation,
[tex]-10(c+3)+4c\ge90[/tex]Resolve the parenthesis first,
[tex]\begin{gathered} -10(c)-10(3)+4c\ge90 \\ -10c-30+4c\ge90 \end{gathered}[/tex]Adding the like terms,
[tex]\begin{gathered} (-10+4)c-30\ge90 \\ -6c-30\ge90 \end{gathered}[/tex]Add 30 both sides,
[tex]\begin{gathered} -6c-30+30\ge90+30 \\ -6c\ge120 \end{gathered}[/tex]Divide both sides by -6, note that the inequality gets reversed when multiplied or divided by a negative number,
[tex]\begin{gathered} \frac{-6c}{-6}\leq\frac{120}{-6} \\ c\leq-20 \end{gathered}[/tex]Thus, the solution set for the given inequation is,
[tex]c\in(-\infty,-20\rbrack[/tex]