Chapter 8 Review Exercises

Chapter Review Exercises

Simplify Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

Exercise $\PageIndex{1}$

$\dfrac{2a+1}{3a−2}$

Answer

$a \ne \dfrac{2}{3}$

Exercise $\PageIndex{2}$

$\dfrac{b−3}{b^2−16}$

Exercise $\PageIndex{3}$

$\dfrac{3xy^2}{5y}$

Answer

$y \ne 0$

Exercise $\PageIndex{4}$

$\dfrac{u−3}{u^2−u−30}$

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

Exercise $\PageIndex{5}$

$\dfrac{4p−1}{p^2+5}$ when $p=−1$

Answer

$−\dfrac{5}{6}$

Exercise $\PageIndex{6}$

$\dfrac{q^2−5}{q+3}$ when $q=7$

Exercise $\PageIndex{7}$

$\dfrac{y^2−8}{y^2−y−2}$ when $y=1$

Answer

$\dfrac{7}{2}$

Example $\PageIndex{8}$

$\dfrac{z^2+2}{4z−z^2}$ when $z=3$

Simplify Rational Expressions

In the following exercises, simplify.

Exercise $\PageIndex{9}$

$\dfrac{10}{24}$

Answer

$\dfrac{5}{12}$

Exercise $\PageIndex{10}$

$\dfrac{8m^4}{16mn^3}$

Exercise $\PageIndex{11}$

$\dfrac{14a−14}{a−1}$

Answer

$14$

Exercise $\PageIndex{12}$

$\dfrac{b^2+7b+12}{b^2+8b+16}$

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

Exercise $\PageIndex{13}$

$\dfrac{c^2−c−2}{4−c^2}$

Answer

$-\dfrac{c+1}{c+2}$

Exercise $\PageIndex{14}$

$\dfrac{d−16}{16−d}$

Exercise $\PageIndex{15}$

$\dfrac{7v−35}{25−v^2}$

Answer

$−\dfrac{7}{5+v}$

Exercise $\PageIndex{16}$

$\dfrac{w^2−3w−28}{49−w^2}$

Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

Exercise $\PageIndex{17}$

$\dfrac{3}{8}·\dfrac{2}{15}$

Answer

$\dfrac{1}{20}$

Exercise $\PageIndex{18}$

$\dfrac{2xy^2}{8y^3}·\dfrac{16y}{24x}$

Exercise $\PageIndex{19}$

$\dfrac{3a^2+21a}{a^2+6a−7}·\dfrac{a−1}{ab}$

Answer

$\dfrac{3}{b}$

Exercise $\PageIndex{20}$

$\dfrac{5z^2}{5z^2+40z+35}·\dfrac{z^2−1}{3z}$

Divide Rational Expressions

In the following exercises, divide.

Exercise $\PageIndex{21}$

$\dfrac{t^2−4t-12}{t^2+8t+12}÷\dfrac{t^2−36}{6t}$

Answer

$\dfrac{6t}{(t+6)^2}$

Exercise $\PageIndex{22}$

$\dfrac{r^2−16}{4}÷\dfrac{r^3−64}{2r^2−8r+32}$

Exercise $\PageIndex{23}$

$\dfrac{11+w}{w−9}÷\dfrac{121−w^2}{9−w}$

Answer

$\dfrac{1}{11+w}$

Exercise $\PageIndex{24}$

$\dfrac{3y^2−12y−63}{4y+3}÷(6y^2−42y)$

Exercise $\PageIndex{25}$

$\dfrac{\dfrac{c^2−64}{3c^2+26c+16}}{\dfrac{c^2−4c−32}{15c+10}}$

Answer

$5c+4$

Exercise $\PageIndex{26}$

$\dfrac{8m^2−8m}{m−4}·\dfrac{m^2+2m−24}{m^2+7m+10}÷\dfrac{2m^2−6m}{m+5}$

​​​​​Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Exercise $\PageIndex{27}$

$\dfrac{3}{5}+\dfrac{2}{5}$

Answer

$1$

Exercise $\PageIndex{28}$

$\dfrac{4a^2}{2a−1}−\dfrac{1}{2a−1}$

Exercise $\PageIndex{29}$

$\dfrac{p^2+10p}{p+5}+\dfrac{25}{p+5}$

Answer

$p+5$

Exercise $\PageIndex{30}$

$\dfrac{3x}{x−1}+\dfrac{2}{x−1}$

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Exercise $\PageIndex{31}$

$\dfrac{d^2}{d+4}−\dfrac{3d+28}{d+4}$

Answer

$d-7$

Exercise $\PageIndex{32}$

$\dfrac{z^2}{z+10}−\dfrac{100}{z+10}$

Exercise $\PageIndex{33}$

$\dfrac{4q^2−q+3}{q^2+6q+5}−\dfrac{3q^2+q+6}{q^2+6q+5}$

Answer

$\dfrac{q−3}{q+5}$

Exercise $\PageIndex{34}$

$\dfrac{5t+4t+3}{t^2−25}−\dfrac{4t^2−8t−32}{t^2−25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

Exercise $\PageIndex{35}$

$\dfrac{18w}{6w−1}+\dfrac{3w−2}{1−6w}$

Answer

$\dfrac{15w+2}{6w−1}$

Exercise $\PageIndex{36}$

$\dfrac{a^2+3a}{a^2−4}−\dfrac{3a−8}{4−a^2}$

Exercise $\PageIndex{37}$

$\dfrac{2b^2+3b−15}{b^2−49}−\dfrac{b^2+16b−1}{49−b^2}$

Answer

$\dfrac{3b−2}{b+7}$

Exercise $\PageIndex{38}$

$\dfrac{8y^2−10y+7}{2y−5}+\dfrac{2y^2+7y+2}{5−2y}$

Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

Exercise $\PageIndex{38}$

$\dfrac{4}{m^2−3m−10},\quad\dfrac{2m}{m^2−m−20}$

Answer

$(m+2)(m−5)(m+4)$

Exercise $\PageIndex{39}$

$\dfrac{6}{n^2−4},\quad\dfrac{2n}{n^2−4n+4}$

Exercise $\PageIndex{40}$

$\dfrac{5}{3p^2+17p−6},\quad\dfrac{2m}{3p^2−23p−8}$

Answer

$(3p+1)(p+6)(p+8)$

Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

Exercise $\PageIndex{41}$

Rewrite as equivalent rational expressions with denominator $(m+2)(m−5)(m+4)$

$\dfrac{4}{m^2−3m−10},\quad\dfrac{2m}{m^2−m−20}$.

Exercise $\PageIndex{42}$

Rewrite as equivalent rational expressions with denominator $(n−2)(n−2)(n+2)$

$\dfrac{6}{n^2−4n+4},\quad\dfrac{2n}{n^2−4}$.

Answer

$\dfrac{6n+12}{(n−2)(n−2)(n+2)},\quad\dfrac{2n^2−4n}{(n−2)(n−2)(n+2)}$

Exercise $\PageIndex{43}$

Rewrite as equivalent rational expressions with denominator $(3p+1)(p+6)(p+8)$

$\dfrac{5}{3p^2+19p+6},\quad\dfrac{7p}{3p^2+25p+8}$

​​​​​​Add Rational Expressions with Different Denominators

In the following exercises, add.

Exercise $\PageIndex{44}$

$\dfrac{2}{3}+\dfrac{3}{5}$

Answer

$\dfrac{19}{15}$

Exercise $\PageIndex{45}$

$\dfrac{7}{5a}+\dfrac{3}{2b}$

Exercise $\PageIndex{46}$

$\dfrac{2}{c−2}+\dfrac{9}{c+3}$

Answer

$\dfrac{11c−12}{(c−2)(c+3)}$

Exercise $\PageIndex{47}$

$\dfrac{3d}{d^2−9}+\dfrac{5}{d^2+6d+9}$

Exercise $\PageIndex{48}$

$\dfrac{2x}{x^2+10x+24}+\dfrac{3x}{x^2+8x+16}$

Answer

$\dfrac{5x^2+26x}{(x+4)(x+4)(x+6)}$

Exercise $\PageIndex{49}$

$\dfrac{5q}{p^{2}q−p^2}+\dfrac{4q}{q^2−1}$

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

Exercise $\PageIndex{50}$

$\dfrac{3v}{v+2}−\dfrac{v+2}{v+8}$

Answer

$\dfrac{2(v^2+10v−2)}{(v+2)(v+8)}$

Exercise $\PageIndex{51}$

$\dfrac{−3w−15}{w^2+w−20}−\dfrac{w+2}{4−w}$

Exercise $\PageIndex{52}$

$\dfrac{7m+3}{m+2}−5$

Answer

$\dfrac{2m−7}{m+2}$

Exercise $\PageIndex{53}$

$\dfrac{n}{n+3}+\dfrac{2}{n−3}−\dfrac{n−9}{n^2−9}$

Exercise $\PageIndex{54}$

$\dfrac{8d}{d^2−64}−\dfrac{4}{d+8}$

Answer

$4d−8$

Exercise $\PageIndex{55}$

$\dfrac{5}{12x^{2}y}+\dfrac{7}{20xy^3}$

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

Exercise $\PageIndex{56}$

$\dfrac{\dfrac{5a}{a+2}}{\dfrac{10a^2}{a^2−4}}$

Answer

$\dfrac{a−2}{2a}$

Exercise $\PageIndex{57}$

$\dfrac{\dfrac{2}{5}+\dfrac{5}{6}}{\dfrac{1}{3}+\dfrac{1}{4}}$

Exercise $\PageIndex{58}$

$\dfrac{x−\dfrac{3x}{x+5}}{\dfrac{1}{x+5}+\dfrac{1}{x−5}}$

Answer

$\dfrac{(x−8)(x−5)}{2}$

Exercise $\PageIndex{59}$

$\dfrac{\dfrac{2}{m}+\dfrac{m}{n}}{\dfrac{n}{m}−\dfrac{1}{n}}$

​​​​​​​Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Exercise $\PageIndex{60}$

$\dfrac{6+\dfrac{2}{q−4}}{\dfrac{5}{q}+4}$

Answer

$\dfrac{(q−2)(q+4)}{5(q−4)}$

Exercise $\PageIndex{61}$

$\dfrac{\dfrac{3}{a^2}−\dfrac{1}{b}}{\dfrac{1}{a}+\dfrac{1}{b^2}}$

Exercise $\PageIndex{62}$

$\dfrac{\dfrac{2}{z^2−49}+\dfrac{1}{z+7}}{\dfrac{9}{z+7}+\dfrac{12}{z−7}}$

Answer

$\dfrac{z−5}{21z+21}$

Exercise $\PageIndex{63}$

$\dfrac{\dfrac{3}{y^2−4y−32}}{\dfrac{2}{y−8}+\dfrac{1}{y+4}}$

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

Exercise $\PageIndex{64}$

$\dfrac{1}{2}+\dfrac{2}{3}=\dfrac{1}{x}$

Answer

$\dfrac{6}{7}$

Exercise $\PageIndex{65}$

$1−\dfrac{2}{m}=\dfrac{8}{m^2}$

Exercise $\PageIndex{66}$

$\dfrac{1}{b−2}+\dfrac{1}{b+2}=\dfrac{3}{b^2−4}$

Answer

$\dfrac{3}{2}$

Exercise $\PageIndex{67}$

$\dfrac{3}{q+8}−\dfrac{2}{q−2}=1$

Exercise $\PageIndex{68}$

$\dfrac{v−15}{v^2−9v+18}=\dfrac{4}{v−3}+\dfrac{2}{v−6}$

Answer

no solution

Exercise $\PageIndex{69}$

$\dfrac{z}{12}+\dfrac{z+3}{3z}=\dfrac{1}{z}$

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Exercise $\PageIndex{70}$

$\dfrac{V}{l}=hw$ for $l$

Answer

$l=\dfrac{V}{hw}$

Exercise $\PageIndex{71}$

$\dfrac{1}{x}−\dfrac{2}{y}=5$ for $y$

Exercise $\PageIndex{72}$

$x=\dfrac{y+5}{z−7}$ for $z$

Answer

$z=\dfrac{y+5+7x}{x}$

Exercise $\PageIndex{73}$

$P=\dfrac{k}{V}$ for $V$

​​​​​​Solve Proportion and Similar Figure Applications Similarity

Solve Proportions

In the following exercises, solve.

Exercise $\PageIndex{74}$

$\dfrac{x}{4}=\dfrac{3}{5}$

Answer

$\dfrac{12}{5}$

Exercise $\PageIndex{75}$

$\dfrac{3}{y}=\dfrac{9}{5}$

Exercise $\PageIndex{76}$

$\dfrac{s}{s+20}=\dfrac{3}{7}$

Answer

$15$

Exercise $\PageIndex{77}$

$\dfrac{t−3}{5}=\dfrac{t+2}{9}$

​​​​​​​In the following exercises, solve using proportions.

Exercise $\PageIndex{78}$

Rachael had a $21$ ounce strawberry shake that has $739$ calories. How many calories are there in a $32$ ounce shake?

Answer

$1161$ calories

Exercise $\PageIndex{79}$

Leo went to Mexico over Christmas break and changed $$525$ dollars into Mexican pesos. At that time, the exchange rate had $$1$ US is equal to $16.25$ Mexican pesos. How many Mexican pesos did he get for his trip?

​​​​​​​Solve Similar Figure Applications

In the following exercises, solve.

Exercise $\PageIndex{80}$

$∆ABC$ is similar to $∆XYZ$. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

Answer

$b=9$; $x=2\dfrac{1}{3}$

Exercise $\PageIndex{81}$

On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is $700$ miles, find the distance from

1. a. Paris to Rome
2. b. Paris to Vienna

Exercise $\PageIndex{82}$

Tony is $5.75$ feet tall. Late one afternoon, his shadow was $8$ feet long. At the same time, the shadow of a nearby tree was $32$ feet long. Find the height of the tree.

Answer

$23$ feet

Exercise $\PageIndex{83}$

The height of a lighthouse in Pensacola, Florida is $150$ feet. Standing next to the statue, $5.5$ foot tall Natalie cast a $1.1$ foot shadow How long would the shadow of the lighthouse be?

​​​​​​​Solve Uniform Motion and Work Applications Problems

Solve Uniform Motion Applications

In the following exercises, solve.

Exercise $\PageIndex{84}$

When making the 5-hour drive home from visiting her parents, Lisa ran into bad weather. She was able to drive $176$ miles while the weather was good, but then driving $10$ mph slower, went $81$ miles in the bad weather. How fast did she drive when the weather was bad?

Answer

45 mph

Exercise $\PageIndex{85}$

Mark is riding on a plane that can fly $490$ miles with a tailwind of $20$ mph in the same time that it can fly $350$ miles against a tailwind of $20$ mph. What is the speed of the plane?​​​​​​​

Exercise $\PageIndex{86}$

John can ride his bicycle $8$ mph faster than Luke can ride his bike. It takes Luke $3$ hours longer than John to ride $48$ miles. How fast can John ride his bike?

Answer

$16$ mph

Exercise $\PageIndex{87}$

Mark was training for a triathlon. He ran $8$ kilometers and biked $32$ kilometers in a total of $3$ hours. His running speed was $8$ kilometers per hour less than his biking speed. What was his running speed?

​​​​​​​Solve Work Applications

In the following exercises, solve.

Exercise $\PageIndex{88}$

Jerry can frame a room in $1$ hour, while Jake takes $4$ hours. How long could they frame a room working together?

Answer

$\dfrac{4}{5}$ hour

Exercise $\PageIndex{89}$

Lisa takes $3$ hours to mow the lawn while her cousin, Barb, takes $2$ hours. How long will it take them working together?

Exercise $\PageIndex{90}$

Jeffrey can paint a house in $6$ days, but if he gets a helper he can do it in $4$ days. How long would it take the helper to paint the house alone?

Answer

$12$ days

Exercise $\PageIndex{91}$

Sue and Deb work together writing a book that takes them $90$ days. If Sue worked alone it would take her $120$ days. How long would it take Deb to write the book alone?

​​​​​​​Use Direct and Inverse Variation

Solve Direct Variation Problems

In the following exercises, solve.

Exercise $\PageIndex{92}$

If $y$ varies directly as $x$, when $y=9$ and $x=3$, find $x$ when $y=21$.

Answer

$7$

Exercise $\PageIndex{93}$

If $y$ varies directly as $x$, when $y=20$ and $x=2$, find $y$ when $x=4$.

Exercise $\PageIndex{94}$

If $m$ varies inversely with the square of $n$, when $m=4$ and $n=6$, find $m$ when $n=2$.

Answer

$36$

Exercise $\PageIndex{95}$

Vanessa is traveling to see her fiancé. The distance, $d$, varies directly with the speed, $v$, she drives. If she travels $258$ miles driving $60$ mph, how far would she travel going $70$ mph?

Exercise $\PageIndex{96}$

If the cost of a pizza varies directly with its diameter, and if an $8$” diameter pizza costs $$12$, how much would a $6$” diameter pizza cost?

Answer

$$9$

Exercise $\PageIndex{97}$

The distance to stop a car varies directly with the square of its speed. It takes $200$ feet to stop a car going $50$ mph. How many feet would it take to stop a car going $60$ mph?

​​​​​​​Solve Inverse Variation Problems

In the following exercises, solve.

Exercise $\PageIndex{98}$

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase $12$ tickets for $$6$, how many tickets can Madelyn afford to buy if the price increased to $$8$?

Answer

$97$ tickets​​​​​​​

Exercise $\PageIndex{99}$

On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an $11$-inch string on a violin has a frequency of $360$ cycles per second, what frequency does a $12$-inch string have?​​​​​​​

Practice Test

In the following exercises, simplify.

Exercise $\PageIndex{1}$

$\dfrac{3a^{2}b}{6ab^2}$

Answer

$\dfrac{a}{2b}$​​​​​​​

Exercise $\PageIndex{2}$

$\dfrac{5b−25}{b^2−25}$

​​​​​​​In the following exercises, perform the indicated operation and simplify.

Exercise $\PageIndex{3}$

$\dfrac{4x}{x+2}·\dfrac{x^2+5x+6}{12x^2}$

Answer

$\dfrac{x+3}{3x}$

Exercise $\PageIndex{4}$

$\dfrac{5y}{4y−8}·\dfrac{y^2−4}{10}$

Exercise $\PageIndex{5}$

$\dfrac{4p}{q}+\dfrac{5}{p}$

Answer

$\dfrac{4+5q}{pq}$

Exercise $\PageIndex{6}$

$\dfrac{1}{z−9}−\dfrac{3}{z+9}$

Exercise $\PageIndex{7}$

$\dfrac{\dfrac{2}{3}+\dfrac{3}{5}}{\dfrac{2}{5}}$

Answer

$\dfrac{19}{16}$

Exercise $\PageIndex{8}$

$\dfrac{\dfrac{1}{m}−\dfrac{1}{n}}{\dfrac{1}{n}+\dfrac{1}{m}}$

In the following exercises, solve each equation.

Exercise $\PageIndex{9}$

$\dfrac{1}{2}+\dfrac{2}{7}=\dfrac{1}{x}$

Answer

$x = \dfrac{14}{11}$

Exercise $\PageIndex{10}$

$\dfrac{5}{y−6}=\dfrac{3}{y+6}$

Exercise $\PageIndex{11}$

$\dfrac{1}{z−5}+\dfrac{1}{z+5}=\dfrac{1}{z^2−25}$

Answer

$z = \dfrac{1}{2}$

Exercise $\PageIndex{12}$

$\dfrac{t}{4}=\dfrac{3}{5}$

Exercise $\PageIndex{13}$

$\dfrac{2}{r−2}=\dfrac{3}{r−1}$

Answer

$r = 4$

In the following exercises, solve.

Exercise $\PageIndex{14}$

If $y$ varies directly with $x$, and $x=5$ when $y=30$, find $x$ when $y=42$.

Exercise $\PageIndex{15}$

If $y$ varies inversely with $x$ and $x=6$ when $y=20$, find $y$ when $x=2$.

Answer

$y=60$

Exercise $\PageIndex{16}$

If $y$ varies inversely with the square of $x$ and $x=3$ when $y=9$, find $y$ when $x=4$.

Exercise $\PageIndex{17}$

The recommended erythromycin dosage for dogs, is $5$ mg for every pound the dog weighs. If Daisy weighs $25$ pounds, how many milligrams of erythromycin should her veterinarian prescribe?

Answer

$125$ mg

Exercise $\PageIndex{18}$

Julia spent $4$ hours Sunday afternoon exercising at the gym. She ran on the treadmill for $10$ miles and then biked for $20$ miles. Her biking speed was $5$ mph faster than her running speed on the treadmill. What was her running speed?

Exercise $\PageIndex{19}$

Kurt can ride his bike for $30$ miles with the wind in the same amount of time that he can go $21$ miles against the wind. If the wind’s speed is $6$ mph, what is Kurt’s speed on his bike?

Answer

$14$ mph

Exercise $\PageIndex{20}$

Amanda jogs to the park $8$ miles using one route and then returns via a $14$-mile route. The return trip takes her $1$ hour longer than her jog to the park. Find her jogging rate.

Exercise $\PageIndex{21}$

An experienced window washer can wash all the windows in Mike’s house in $2$ hours, while a new trainee can wash all the windows in $7$ hours. How long would it take them working together?

Answer

$1\frac{5}{9}$ hour

Exercise $\PageIndex{22}$

Josh can split a truckload of logs in $8$ hours, but working with his dad they can get it done in $3$ hours. How long would it take Josh’s dad working alone to split the logs?

Exercise $\PageIndex{23}$

The price that Tyler pays for gas varies directly with the number of gallons he buys. If $24$ gallons cost him $$59.76$, what would $30$ gallons cost?

Answer

$$74.70$

Exercise $\PageIndex{24}$

The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of $29.5$ liters with $2000$ psi, what is the volume if the tank has a $14.7$ psi rating? Round to the nearest whole number.

Exercise $\PageIndex{25}$

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio, as shown on the figure below, that gives the map distances between these cities in inches.

The actual distance from Dayton to Cincinnati is $48$ miles. What is the actual distance between Dayton and Columbus?

Answer

$64$ miles

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