Let x be the number of weeks passed.
Denote by J(x) the money with Joe.
Denote by B(x) the money with his brother.
Denote by s(x) the money they raised together.
[tex]J(x)=assets+money \: raised \\ J(x)=170 + 20x[/tex]
[tex]B(x)=assets+money \: raised \\ B(x)=80+40x[/tex]
If we want Joe's brother to have more money at the x week, then B(x) has to be greater than J(x)
[tex]B(x) > J(x) \\ 80 + 40x > 170 + 20x \\ 40x - 20x > 170 - 80 \\ 20x > 90 \\ x > \frac{90}{20} \\ x > 4.5 \: \: \: so \: \: \: x = 5[/tex]
We know that at the 5th week, Joe's brother will have raised more money than Joe, but how much money did they raise together by then?
[tex]s(x) = B(x) + J(x) \\ s(5) = B(5) + J(5) \\ s(5) = 80 + 40(5) + 170 + 20(5) \\ s(5) = 80 + 200 + 170 + 100 \\ s(5) = 550 \: dollars \: raised[/tex]
