8.4 Add and subtract rational expressions with unlike denominators  (Page 4/4)

$\frac{5}{x^2-2x-8},\frac{2x}{x^2-x-12}$

$\frac{8}{y^2+12y+35},\frac{3y}{y^2+y-42}$

$\frac{9}{z^2+2z-8},\frac{4z}{z^2-4}$

$\frac{6}{a^2+14a+45},\frac{5a}{a^2-81}$

$\frac{4}{b^2+6b+9},\frac{2b}{b^2-2b-15}$

$\frac{5}{c^2-4c+4},\frac{3c}{c^2-10c+16}$

$\frac{2}{3d^2+14d-5},\frac{5d}{3d^2-19d+6}$

$\frac{3}{5m^2-3m-2},\frac{6m}{5m^2+17m+6}$

$\frac{5}{x^2-2x-8},\frac{2x}{x^2-x-12}$
LCD $(x-4)(x+2)(x+3)$

$\frac{8}{y^2+12y+35},\frac{3y}{y^2+y-42}$
LCD $(y+7)(y+5)(y-6)$

$\frac{9}{z^2+2z-8},\frac{4z}{z^2-4}$
LCD $(z-2)(z+4)(z+2)$

$\frac{6}{a^2+14a+45},\frac{5a}{a^2-81}$
LCD $(a+9)(a+5)(a-9)$

$\frac{4}{b^2+6b+9},\frac{2b}{b^2-2b-15}$
LCD $(b+3)(b+3)(b-5)$

$\frac{5}{c^2-4c+4},\frac{3c}{c^2-10c+10}$
LCD $(c-2)(c-2)(c-8)$

$\frac{2}{3d^2+14d-5},\frac{5d}{3d^2-19d+6}$
LCD $(3d-1)(d+5)(d-6)$

$\frac{3}{5m^2-3m-2},\frac{6m}{5m^2+17m+6}$
LCD $(5m+2)(m-1)(m+3)$

$\frac{5}{24}+\frac{11}{36}$

$\frac{7}{30}+\frac{13}{45}$

$\frac{9}{20}+\frac{11}{30}$

$\frac{8}{27}+\frac{7}{18}$

$\frac{7}{10x^{2}y}+\frac{4}{15x{y}^2}$

$\frac{1}{12a^3{b}^2}+\frac{5}{9a^2{b}^3}$

$\frac{1}{2m}+\frac{7}{8m^{2}n}$

$\frac{5}{6p^{2}q}+\frac{1}{4p}$

$\frac{3}{r+4}+\frac{2}{r-5}$

$\frac{4}{s-7}+\frac{5}{s+3}$

$\frac{8}{t+5}+\frac{6}{t-5}$

$\frac{7}{v+5}+\frac{9}{v-5}$

$\frac{5}{3w-2}+\frac{2}{w+1}$

$\frac{4}{2x+5}+\frac{2}{x-1}$

$\frac{2y}{y+3}+\frac{3}{y-1}$

$\frac{3z}{z-2}+\frac{1}{z+5}$

$\frac{5b}{a^{2}b-2a^2}+\frac{2b}{b^2-4}$

$\frac{4}{cd+3c}+\frac{1}{d^2-9}$

$\frac{2m}{3m-3}+\frac{5m}{m^2+3m-4}$

$\frac{3}{4n+4}+\frac{6}{n^2-n-2}$

$\frac{3}{n^2+3n-18}+\frac{4n}{n^2+8n+12}$

$\frac{6}{q^2-3q-10}+\frac{5q}{q^2-8q+15}$

$\frac{3r}{r^2+7r+6}+\frac{9}{r^2+4r+3}$

$\frac{2s}{s^2+2s-8}+\frac{4}{s^2+3s-10}$

$\frac{t}{t-6}-\frac{t-2}{t+6}$

$\frac{v}{v-3}-\frac{v-6}{v+1}$

$\frac{w+2}{w+4}-\frac{w}{w-2}$

$\frac{x-3}{x+6}-\frac{x}{x+3}$

$\frac{y-4}{y+1}-\frac{1}{y+7}$

$\frac{z+8}{z-3}-\frac{z}{z-2}$

$\frac{5a}{a+3}-\frac{a+2}{a+6}$

$\frac{3b}{b-2}-\frac{b-6}{b-8}$

$\frac{6c}{c^2-25}-\frac{3}{c+5}$

$\frac{4d}{d^2-81}-\frac{2}{d+9}$

$\frac{6}{m+6}-\frac{12m}{m^2-36}$

$\frac{4}{n+4}-\frac{8n}{n^2-16}$

$\frac{\mathrm{-9}p-17}{p^2-4p-21}-\frac{p+1}{7-p}$

$\frac{7q+8}{q^2-2q-24}-\frac{q+2}{4-q}$

$\frac{\mathrm{-2}r-16}{r^2+6r-16}-\frac{5}{2-r}$

$\frac{2t-30}{t^2+6t-27}-\frac{2}{3-t}$

$\frac{5v-2}{v+3}-4$

$\frac{6w+5}{w-1}+2$

$\frac{2x+7}{10x-1}+3$

$\frac{8y-4}{5y+2}-6$

$\frac{5a}{a-2}+\frac{9}{a}-\frac{2a+18}{a^2-2a}$

$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2b^2-10b}$

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{11c}{c^2-4}$

$\frac{6d}{d-5}+\frac{1}{d+4}-\frac{7d-5}{d^2-d-20}$

$\frac{6a}{3ab+b^2}+\frac{3a}{9a^2-b^2}$

$\frac{2c}{2c+10}+\frac{7c}{c^2+9c+20}$

$\frac{6d}{d^2-64}-\frac{3}{d-8}$

$\frac{5}{n+7}-\frac{10n}{n^2-49}$

$\frac{4m}{m^2+6m-7}+\frac{2}{m^2+10m+21}$

$\frac{3p}{p^2+4p-12}+\frac{1}{p^2+p-30}$

$\frac{\mathrm{-5}n-5}{n^2+n-6}+\frac{n+1}{2-n}$

$\frac{\mathrm{-4}b-24}{b^2+b-30}+\frac{b+7}{5-b}$

$\frac{7}{15p}+\frac{5}{18pq}$

$\frac{3}{20a^2}+\frac{11}{12a{b}^2}$

$\frac{4}{x-2}+\frac{3}{x+5}$

$\frac{6}{m+4}+\frac{9}{m-8}$

$\frac{2q+7}{y+4}-2$

$\frac{3y-1}{y+4}-2$

$\frac{z+2}{z-5}-\frac{z}{z+1}$

$\frac{t}{t-5}-\frac{t-1}{t+5}$

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{d^2+2d}$

$\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{q^2+2q-15}$

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in $t$ hours, so in 1 hour she can decorate $\frac{1}{t}$ of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate $\frac{1}{t+3}$ of the cupcakes.

1. ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions $\frac{1}{t}+\frac{1}{t+3}.$
2. ⓑ Evaluate your answer to part (a) when $t=5.$

Kayaking When Trina kayaks upriver, it takes her $\frac{5}{3-c}$ hours to go 5 miles, where $c$ is the speed of the river current. It takes her $\frac{5}{3+c}$ hours to kayak 5 miles down the river.

1. ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding $\frac{5}{3-c}+\frac{5}{3+c}.$
2. ⓑ Evaluate your answer to part (a) when $c=1$ to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.

Felipe thinks $\frac{1}{x}+\frac{1}{y}$ is $\frac{2}{x+y}.$

1. ⓐ Choose numerical values for x and y and evaluate $\frac{1}{x}+\frac{1}{y}.$
2. ⓑ Evaluate $\frac{2}{x+y}$ for the same values of x and y you used in part (a).
3. ⓒ Explain why Felipe is wrong.
4. ⓓ Find the correct expression for $\frac{1}{x}+\frac{1}{y}.$

Simplify the expression $\frac{4}{n^2+6n+9}-\frac{1}{n^2-9}$ and explain all your steps.