The average of the three grades is given by:
[tex]average=\frac{84+95+x}{3}[/tex]
where x is the grade of the final exam.
(a) So, in order to have an average of at least 90, the grade x must be:
[tex]\begin{gathered} \frac{84+95+x}{3}=90 \\ \\ 84+95+x=90\cdot3 \\ \\ 179+x=270 \\ \\ x=270-179 \\ \\ x=91 \end{gathered}[/tex]
Therefore, to earn an A in the course, you must get at least 91 on the final.
(b) If the final average is less than 80, we have:
[tex]\begin{gathered} \frac{84+95+x}{3}<80 \\ \\ 84+95+x<80\cdot3 \\ \\ 179+x<240 \\ \\ x<240-179 \\ \\ x<61 \end{gathered}[/tex]
Therefore, the grades on the final that will cause this to happen are the grades that are less than 61.
