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${\left(x-7\right)}^{4}\xf7\frac{{\left(x-7\right)}^{3}}{x+1}$
$\left(x-7\right)\left(x+1\right)$
${\left(4x+9\right)}^{6}\xf7\frac{{\left(4x+9\right)}^{2}}{{\left(3x+1\right)}^{4}}$
$5x+\frac{2{x}^{2}+1}{x-4}$
$\frac{7{x}^{2}-20x+1}{\left(x-4\right)}$
$2y+\frac{4{y}^{2}+5}{y-1}$
$\frac{{y}^{2}+4y+4}{{y}^{2}+10y+21}\xf7\left(y+2\right)$
$\frac{\left(y+2\right)}{\left(y+3\right)\left(y+7\right)}$
$2x-3+\frac{4{x}^{2}+x-1}{x-1}$
$\frac{3x+1}{{x}^{2}+3x+2}+\frac{5x+6}{{x}^{2}+6x+5}-\frac{3x-7}{{x}^{2}-2x-35}$
$\frac{5{x}^{3}-26{x}^{2}-192x-105}{\left({x}^{2}-2x-35\right)\left(x+1\right)\left(x+2\right)}$
$\frac{5a+3b}{8{a}^{2}+2ab-{b}^{2}}-\frac{3a-b}{4{a}^{2}-9ab+2{b}^{2}}-\frac{a+5b}{4{a}^{2}+3ab-{b}^{2}}$
$\frac{3{x}^{2}+6x+10}{10{x}^{2}+11x-6}+\frac{2{x}^{2}-4x+15}{2{x}^{2}-11x-21}$
$\frac{13{x}^{3}-39{x}^{2}+51x-100}{\left(2x+3\right)\left(x-7\right)\left(5x-2\right)}$
For the following problems, solve the rational equations.
$\frac{4x}{5}+\frac{3x-1}{15}=\frac{29}{25}$
$\frac{5x-1}{6}+\frac{3x+4}{9}=\frac{-8}{9}$
$\frac{4}{x-1}+\frac{7}{x+2}=\frac{43}{{x}^{2}+x-2}$
$\frac{-5}{y-3}+\frac{2}{y-3}=\frac{3}{y-3}$
$\frac{2m+5}{m-8}+\frac{9}{m-8}=\frac{30}{m-8}$
No solution; $m=8$ is excluded.
$\frac{r+6}{r-1}-\frac{3r+2}{r-1}=\frac{-6}{r-1}$
$\frac{8b+1}{b-7}-\frac{b+5}{b-7}=\frac{45}{b-7}$
No solution; $b=7$ is excluded.
Solve $z=\frac{x-\overline{x}}{x}\text{\hspace{0.17em}for\hspace{0.17em}}s.$
Solve $A=P\left(1+rt\right)\text{\hspace{0.17em}for\hspace{0.17em}}t.$
$t=\frac{A-P}{Pr}$
Solve $\frac{1}{R}=\frac{1}{E}+\frac{1}{F}\text{\hspace{0.17em}for\hspace{0.17em}}E.$
Solve $Q=\frac{2mn}{s+t}\text{\hspace{0.17em}for\hspace{0.17em}}t.$
$t=\frac{2mn-Qs}{Q}$
Solve $I=\frac{E}{R+r}\text{\hspace{0.17em}for\hspace{0.17em}}r.$
For the following problems, find the solution.
When the same number is subtracted from both terms of the fraction $\frac{7}{12},$ the result is $\frac{1}{2}.$ What is the number?
2
When the same number is added to both terms of the fraction $\frac{13}{15},$ the result is $\frac{8}{9}.$ What is the number?
When three fourths of a number is added to the reciprocal of the number, the result is $\frac{173}{16}.$ What is the number?
No rational solution.
When one third of a number is added to the reciprocal of the number, the result is $\frac{-127}{90}.$ What is the number?
Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?
$3\frac{15}{16}\text{\hspace{0.17em}hrs}$
Debbie can complete an algebra assignment in $\frac{3}{4}$ of an hour. Sandi, who plays her radio while working, can complete the same assignment in $1\frac{1}{4}$ hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?
An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?
24 hrs
Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?
The pressure due to surface tension in a spherical bubble is given by
$P=\frac{4T}{r},$ where
$T$ is the surface tension of the liquid, and
$r$ is the radius of the bubble.
(a) Determine the pressure due to surface tension within a soap bubble of radius
$\frac{1}{2}$ inch and surface tension 22.
(b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.
(a) 176 units of pressure; (b) $\frac{5}{4}$ units of length
The equation
$\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$ relates an objects distance
$p$ from a lens and the image distance
$q$ from the lens to the focal length
$f$ of the lens.
(a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of
$1\frac{7}{8}$ cm and the object distance is 3 cm away from the lens.
For the following problems, divide the polynomials.
${c}^{2}+3c-88$ by $c-8$
${y}^{3}-2{y}^{2}-49y-6$ by $y+6$
${m}^{4}+2{m}^{3}-8{m}^{2}-m+2$ by $m-2$
${m}^{3}+4{m}^{2}-1$
$3{r}^{2}-17r-27$ by $r-7$
${a}^{3}-3{a}^{2}-56a+10$ by $a-9$
${a}^{2}+6a-2-\frac{8}{a-9}$
${x}^{3}-x+1$ by $x+3$
$5{x}^{6}+5{x}^{5}-2{x}^{4}+5{x}^{3}-7{x}^{2}-8x+6$ by ${x}^{2}+x-1$
$-4{b}^{7}-3{b}^{6}-22{b}^{5}-19{b}^{4}+12{b}^{3}-6{b}^{2}+b+4$ by ${b}^{2}+6$
${a}^{4}+6{a}^{3}+4{a}^{2}+12a+8$ by ${a}^{2}+3a+2$
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