유리함수

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

In the following exercises, simplify.

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

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.

• ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions $\frac{1}{t}+\frac{1}{t+3}.$
• ⓑ 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.

• ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding $\frac{5}{3-c}+\frac{5}{3+c}.$
• ⓑ 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 $\frac{2}{x+y}.$

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

Simplify: $\frac{\frac{3}{5}}{\frac{9}{10}}.$

Simplify: $\frac{1-\frac{1}{3}}{{4}^{2}+4\cdot 5}.$