Question

What are the absolute value answers to 5n+24=|8-3n|?

Answer 1

$5n+24=|8-3n|\\ 5n+24\geq0\\ 5n\geq-24\\n\geq-\dfrac{24}{5}\\ 8-3n=5n+24 \vee 8-3n=-5n-24\\ 8n=-16 \vee 2n=-32\\ n=-2 \vee n=-16\\\\ -16\not\geq-\dfrac{24}{5} \Rightarrow \boxed{n=-2}$

Answer 2

$\hbox{the domain:} \\ 5n+24 \geq 0 \\ 5n \geq -24 \\ n \geq -\frac{24}{5} \\ n \geq -4 \frac{4}{5}$

$5n+24=|8-3n| \\ 5n+24=8-3n \ \lor \ -(5n+24)=8-3n \\ 5n+3n=8-24 \ \lor \ -5n-24=8-3n \\ 8n=-16 \ \lor \ -5n+3n=8+24 \\ n=-2 \ \lor \ -2n=32 \\ n=-2 \ \lor \ n=-16$

$n=-2 \geq -4 \frac{4}{5} \hbox{ - it is a solution} \\ n=-16 < -4 \frac{4}{5} \hbox{ - it isn't a solution} \\ \\ \boxed{n=-2}$