EDIT: One more problem I'm having with fuctions...
solve:
f(2/3) = (x^2+3)/(x-5)
having a hard time here
I'll do the second rational equation ... you try the first one after I show you how ...
$\displaystyle \frac{x+10}{x+3} = \frac{x+8}{x+5}$
cross multiply ...
$\displaystyle (x+10)(x+5) = (x+3)(x+8)$
expand (multiply out) both sides ...
$\displaystyle x^2 + 15x + 50 = x^2 + 11x + 24$
combine like terms ...
$\displaystyle 4x = -26$
$\displaystyle x = -\frac{13}{2}$
here's the first one ... I'll leave the easier one for you.
$\displaystyle \sqrt{x^2 + 16} = 3+x$
square both sides ...
$\displaystyle x^2 + 16 = 9 + 6x + x^2$
combine like terms ...
$\displaystyle 7 = 6x$
$\displaystyle \frac{7}{6} = x$
important! ... check the solution
$\displaystyle \sqrt{\frac{49}{36} + 16} = 3 + \frac{7}{6}
$
$\displaystyle \sqrt{\frac{625}{36}} = \frac{25}{6}
$
$\displaystyle \frac{25}{6} = \frac{25}{6}$
checks good.
$\displaystyle \frac{(\frac{2}{3})^2 + 3}{\frac{2}{3}-5}$
$\displaystyle \frac{(\frac{4}{9}) + 3}{\frac{2}{3}-5}$
$\displaystyle \frac{\frac{4+27}{9}}{\frac{2-15}{3}}$
$\displaystyle \frac{\frac{31}{9}}{\frac{-13}{3}}$
$\displaystyle \frac{(3)(31)}{(9)(-13)}$
$\displaystyle -\frac{31}{39}$