## Introductory Algebra for College Students (7th Edition)

$\displaystyle \frac{y+3}{3y+8}$
To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, factor and simplify the result. --- Don't forget to place the second numerator in parentheses when subtracting. $\displaystyle \frac{4y^{2}+5}{9y^{2}-64}-\frac{y^{2}-y+29}{9y^{2}-64}= \displaystyle \frac{4y^{2}+5-y^{2}-y+29)}{9y^{2}-64}$ $= \displaystyle \frac{4y^{2}+5-y^{2}+y-29}{9y^{2}-64}$ $= \displaystyle \frac{3y^{2}+y-24}{9y^{2}-64}\qquad$... factor what we can... ... Numerator: $...$ two factors of $ac=-72$ whose sum is 1... are $-8$ and $+9$. ... Rewrite bx and factor in pairs: $3y^{2}+y-24=3y^{2}+9y-8y-24$ $=3y(y+3)-8(y+3)=(y+3)(3y-8)$ ... Denominator: a difference of squares, $(3y)^{2}-8^{2}$ $=\displaystyle \frac{(y+3)(3y-8)}{(3y+8)(3y-8)}$ ... reduce (divide the numerator and denominator with common factors) $=\displaystyle \frac{y+3}{3y+8}$