### Review Exercises

##### Simplify, Multiply, and Divide 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.

$\frac{b-7}{{b}^{2}-25}$

$\frac{x-3}{{x}^{2}-x-30}$

**Simplify Rational Expressions**

In the following exercises, simplify.

$\frac{9{m}^{4}}{18m{n}^{3}}$

$\frac{7v-35}{25-{v}^{2}}$

**Multiply Rational Expressions**

In the following exercises, multiply.

$\frac{3x{y}^{2}}{8{y}^{3}}\xb7\frac{16{y}^{2}}{24x}$

$\frac{2{y}^{2}+y-3}{4-{y}^{2}}\xb7\frac{{y}^{2}-4y+4}{2{y}^{2}+11y+12}$

**Divide Rational Expressions**

In the following exercises, divide.

$\frac{{y}^{2}-16}{4}\xf7\frac{{y}^{3}-64}{2{y}^{2}+8y+32}$

$\frac{3{y}^{2}-12y-63}{4y+3}\xf7(6{y}^{2}-42y)$

$\frac{8{a}^{2}+16a}{a-4}\xb7\frac{{a}^{2}+2a-24}{{a}^{2}+7a+10}\xf7\frac{2{a}^{2}-6a}{a+5}$

**Multiply and Divide Rational Functions**

Find $R\left(x\right)=f\left(x\right)\xb7g\left(x\right)$ where $f\left(x\right)=\frac{9{x}^{2}+9x}{{x}^{2}-3x-4}$ and $g\left(x\right)=\frac{{x}^{2}-16}{3{x}^{2}+12x}.$

Find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ where $f(x)=\frac{27{x}^{2}}{3x-21}$ and

$g\left(x\right)=\frac{9{x}^{2}+54x}{{x}^{2}-x-42}.$

##### Add and Subtract Rational Expressions

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

In the following exercises, perform the indicated operations.

$\frac{4{a}^{2}}{2a-1}-\frac{1}{2a-1}$

$\frac{7{x}^{2}}{{x}^{2}-9}+\frac{21x}{{x}^{2}-9}$

$\frac{{y}^{2}}{y+11}-\frac{121}{y+11}$

$\frac{5{t}^{2}+4t+3}{{t}^{2}-25}-\frac{4{t}^{2}-8t-32}{{t}^{2}-25}$

**Add and Subtract Rational Expressions Whose Denominators Are Opposites**

In the following exercises, add and subtract.

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

$\frac{8{y}^{2}-10y+7}{2y-5}+\frac{2{y}^{2}+7y+2}{5-2y}$

**Find the Least Common Denominator of Rational Expressions**

In the following exercises, find the LCD.

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

**Add and Subtract Rational Expressions with Unlike Denominators**

In the following exercises, perform the indicated operations.

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

$\frac{3x}{{x}^{2}-9}+\frac{5}{{x}^{2}+6x+9}$

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

$\frac{\mathrm{-3}w-15}{{w}^{2}+w-20}-\frac{w+2}{4-w}$

$\frac{n}{n+3}+\frac{2}{n-3}-\frac{n+9}{{n}^{2}-9}$

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

**Add and Subtract Rational Functions**

In the following exercises, find $R\left(x\right)=f\left(x\right)+g\left(x\right)$ where $f\left(x\right)$ and $g\left(x\right)$ are given.

$f\left(x\right)=\frac{\mathrm{-4}x+31}{{x}^{2}+x-30},$$g\left(x\right)=\frac{5}{x+6}$

In the following exercises, find $R\left(x\right)=f\left(x\right)-g\left(x\right)$ where $f\left(x\right)$ and $g\left(x\right)$ are given.

$f\left(x\right)=\frac{7}{x+6},$$g\left(x\right)=\frac{14x}{{x}^{2}-36}$

##### Simplify Complex Rational Expressions

**Simplify a Complex Rational Expression by Writing It as Division**

In the following exercises, simplify.

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

$\frac{\frac{2}{m}+\frac{m}{n}}{\frac{n}{m}-\frac{1}{n}}$

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

In the following exercises, simplify.

$\frac{\frac{3}{{a}^{2}}-\frac{1}{b}}{\frac{1}{a}+\frac{1}{{b}^{2}}}$

$\frac{\frac{3}{{y}^{2}-4y-32}}{\frac{2}{y-8}+\frac{1}{y+4}}$

##### 7.4 Solve Rational Equations

**Solve Rational Equations**

In the following exercises, solve.

$1-\frac{2}{m}=\frac{8}{{m}^{2}}$

$\frac{3}{q+8}-\frac{2}{q-2}=1$

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

**Solve Rational Equations that Involve Functions**

For rational function, $f\left(x\right)=\frac{x+2}{{x}^{2}-6x+8},$ ⓐ find the domain of the function ⓑ solve $f\left(x\right)=1$ ⓒ find the points on the graph at this function value.

For rational function, $f\left(x\right)=\frac{2-x}{{x}^{2}+7x+10},$ ⓐ find the domain of the function ⓑ solve $f\left(x\right)=2$ ⓒ find the points on the graph at this function value.

**Solve a Rational Equation for a Specific Variable**

In the following exercises, solve for the indicated variable.

$\frac{1}{x}-\frac{2}{y}=5$ for $y.$

$P=\frac{k}{V}$ for $V.$

##### Solve Applications with Rational Equations

**Solve Proportions**

In the following exercises, solve.

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

$\frac{t-3}{5}=\frac{t+2}{9}$

**Solve Using Proportions**

In the following exercises, solve.

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

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.

$\text{\Delta}ABC$ is similar to $\text{\Delta}XYZ.$ The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

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

ⓐ Paris to Rome

ⓑ Paris to Vienna

Francesca is 5.75 feet tall. Late one afternoon, her 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.

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

**Solve Uniform Motion Applications**

In the following exercises, solve.

When making the 5-hour drive home from visiting her parents, Lolo 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 when it turned bad. How fast did she drive when the weather was bad?

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?

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

Curtis 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.

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

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

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?

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

**Solve Direct Variation Problems**

In the following exercises, solve.

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

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?

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?

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.

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

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?

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?

##### Solve Rational Inequalities

**Solve Rational Inequalities**

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{5x}{x-2}>1$

$\frac{1}{{x}^{2}-4x-12}<0$

$\frac{4}{x-2}<\frac{3}{x+1}$

**Solve an Inequality with Rational Functions**

In the following exercises, solve each rational function inequality and write the solution in interval notation

Given the function, $R\left(x\right)=\frac{x-5}{x-2},$ find the values of $x$ that make the function greater than or equal to 0.

Given the function, $R\left(x\right)=\frac{x+1}{x+3},$ find the values of $x$ that make the function less than or equal to 0.

The function

$C(x)=150x+\mathrm{100,000}$ represents the cost to produce $x,$ number of items. Find ⓐ the average cost function, $c(x)$ ⓑ how many items should be produced so that the average cost is less than $160.

Tillman is starting his own business by selling tacos at the beach. Accounting for the cost of his food truck and ingredients for the tacos, the function $C(x)=2x+\mathrm{6,000}$ represents the cost for Tillman to produce $x,$ tacos. Find ⓐ the average cost function, $c(x)$ for Tillman’s Tacos ⓑ how many tacos should Tillman produce so that the average cost is less than $4.