# Chapter 7 Review Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## 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 $$\PageIndex{1}$$

$$\dfrac{2a+1}{3a−2}$$

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

##### Exercise $$\PageIndex{2}$$

$$\dfrac{b−3}{b^2−16}$$

##### Exercise $$\PageIndex{3}$$

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

$$y \ne 0$$

##### 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 $$p=−1$$

$$−\dfrac{5}{6}$$

##### Exercise $$\PageIndex{6}$$

$$\dfrac{q^2−5}{q+3}$$ when $$q=7$$

##### Exercise $$\PageIndex{7}$$

$$\dfrac{y^2−8}{y^2−y−2}$$ when $$y=1$$

$$\dfrac{7}{2}$$

##### Example $$\PageIndex{8}$$

$$\dfrac{z^2+2}{4z−z^2}$$ when $$z=3$$

Simplify Rational Expressions

In the following exercises, simplify.

##### Exercise $$\PageIndex{9}$$

$$\dfrac{10}{24}$$

$$\dfrac{5}{12}$$

##### Exercise $$\PageIndex{10}$$

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

##### Exercise $$\PageIndex{11}$$

$$\dfrac{14a−14}{a−1}$$

$$14$$

##### Exercise $$\PageIndex{12}$$

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

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

##### Exercise $$\PageIndex{13}$$

$$\dfrac{c^2−c−2}{4−c^2}$$

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

##### Exercise $$\PageIndex{14}$$

$$\dfrac{d−16}{16−d}$$

##### Exercise $$\PageIndex{15}$$

$$\dfrac{7v−35}{25−v^2}$$

$$−\dfrac{7}{5+v}$$

##### 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 $$\PageIndex{17}$$

$$\dfrac{3}{8}·\dfrac{2}{15}$$

$$\dfrac{1}{20}$$

##### Exercise $$\PageIndex{18}$$

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

##### Exercise $$\PageIndex{19}$$

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

$$\dfrac{3}{b}$$

##### Exercise $$\PageIndex{20}$$

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

Divide Rational Expressions

In the following exercises, divide.

##### Exercise $$\PageIndex{21}$$

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

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

##### Exercise $$\PageIndex{22}$$

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

##### Exercise $$\PageIndex{23}$$

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

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

##### Exercise $$\PageIndex{24}$$

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

##### Exercise $$\PageIndex{25}$$

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

$$5c+4$$

##### 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

##### Exercise $$\PageIndex{27}$$

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

$$1$$

##### Exercise $$\PageIndex{28}$$

$$\dfrac{4a^2}{2a−1}−\dfrac{1}{2a−1}$$

##### Exercise $$\PageIndex{29}$$

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

$$p+5$$

##### Exercise $$\PageIndex{30}$$

$$\dfrac{3x}{x−1}+\dfrac{2}{x−1}$$

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

##### Exercise $$\PageIndex{31}$$

$$\dfrac{d^2}{d+4}−\dfrac{3d+28}{d+4}$$

$$d-7$$

##### Exercise $$\PageIndex{32}$$

$$\dfrac{z^2}{z+10}−\dfrac{100}{z+10}$$

##### Exercise $$\PageIndex{33}$$

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

$$\dfrac{q−3}{q+5}$$

##### 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 $$\PageIndex{35}$$

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

$$\dfrac{15w+2}{6w−1}$$

##### Exercise $$\PageIndex{36}$$

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

##### Exercise $$\PageIndex{37}$$

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

$$\dfrac{3b−2}{b+7}$$

##### 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 $$\PageIndex{38}$$

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

$$(m+2)(m−5)(m+4)$$

##### Exercise $$\PageIndex{39}$$

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

##### Exercise $$\PageIndex{40}$$

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

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

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 $$(n−2)(n−2)(n+2)$$

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

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

##### 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

##### Exercise $$\PageIndex{44}$$

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

$$\dfrac{19}{15}$$

##### Exercise $$\PageIndex{45}$$

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

##### Exercise $$\PageIndex{46}$$

$$\dfrac{2}{c−2}+\dfrac{9}{c+3}$$

$$\dfrac{11c−12}{(c−2)(c+3)}$$

##### Exercise $$\PageIndex{47}$$

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

##### Exercise $$\PageIndex{48}$$

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

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

##### 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 $$\PageIndex{50}$$

$$\dfrac{3v}{v+2}−\dfrac{v+2}{v+8}$$

$$\dfrac{2(v^2+10v−2)}{(v+2)(v+8)}$$

##### Exercise $$\PageIndex{51}$$

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

##### Exercise $$\PageIndex{52}$$

$$\dfrac{7m+3}{m+2}−5$$

$$\dfrac{2m−7}{m+2}$$

##### Exercise $$\PageIndex{53}$$

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

##### Exercise $$\PageIndex{54}$$

$$\dfrac{8d}{d^2−64}−\dfrac{4}{d+8}$$

$$4d−8$$

##### 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 $$\PageIndex{56}$$

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

$$\dfrac{a−2}{2a}$$

##### Exercise $$\PageIndex{57}$$

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

##### Exercise $$\PageIndex{58}$$

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

$$\dfrac{(x−8)(x−5)}{2}$$

##### 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 $$\PageIndex{60}$$

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

$$\dfrac{(q−2)(q+4)}{5(q−4)}$$

##### Exercise $$\PageIndex{61}$$

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

##### Exercise $$\PageIndex{62}$$

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

$$\dfrac{z−5}{21z+21}$$

##### 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 $$\PageIndex{64}$$

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

$$\dfrac{6}{7}$$

##### Exercise $$\PageIndex{65}$$

$$1−\dfrac{2}{m}=\dfrac{8}{m^2}$$

##### Exercise $$\PageIndex{66}$$

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

$$\dfrac{3}{2}$$

##### Exercise $$\PageIndex{67}$$

$$\dfrac{3}{q+8}−\dfrac{2}{q−2}=1$$

##### Exercise $$\PageIndex{68}$$

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

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 $$l$$

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

##### Exercise $$\PageIndex{71}$$

$$\dfrac{1}{x}−\dfrac{2}{y}=5$$ for $$y$$

##### Exercise $$\PageIndex{72}$$

$$x=\dfrac{y+5}{z−7}$$ for $$z$$

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

##### Exercise $$\PageIndex{73}$$

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

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

Solve Proportions

In the following exercises, solve.

##### Exercise $$\PageIndex{74}$$

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

$$\dfrac{12}{5}$$

##### Exercise $$\PageIndex{75}$$

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

##### Exercise $$\PageIndex{76}$$

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

$$15$$

##### 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?

$$1161$$ 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.

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

##### 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.

$$23$$ 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?

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?

$$16$$ 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?

$$\dfrac{4}{5}$$ 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?

$$12$$ 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 $$y=21$$.

$$7$$

##### 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 $$n=2$$.

$$36$$

##### 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?

$$9$$

##### 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 $$8$$?

$$97$$ 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 $$\PageIndex{1}$$

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

$$\dfrac{a}{2b}$$​​​​​​​

##### Exercise $$\PageIndex{2}$$

$$\dfrac{5b−25}{b^2−25}$$

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

##### Exercise $$\PageIndex{3}$$

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

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

##### Exercise $$\PageIndex{4}$$

$$\dfrac{5y}{4y−8}·\dfrac{y^2−4}{10}$$

##### Exercise $$\PageIndex{5}$$

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

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

##### Exercise $$\PageIndex{6}$$

$$\dfrac{1}{z−9}−\dfrac{3}{z+9}$$

##### Exercise $$\PageIndex{7}$$

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

$$\dfrac{19}{16}$$

##### Exercise $$\PageIndex{8}$$

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

In the following exercises, solve each equation.

##### Exercise $$\PageIndex{9}$$

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

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

##### Exercise $$\PageIndex{10}$$

$$\dfrac{5}{y−6}=\dfrac{3}{y+6}$$

##### Exercise $$\PageIndex{11}$$

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

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

##### Exercise $$\PageIndex{12}$$

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

##### Exercise $$\PageIndex{13}$$

$$\dfrac{2}{r−2}=\dfrac{3}{r−1}$$

$$r = 4$$

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 $$x=2$$.

$$y=60$$

##### 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?

$$125$$ 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?

$$14$$ 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?

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

$$74.70$$

##### 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?

$$64$$ miles