### 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.

$\frac{b-3}{{b}^{2}-16}$

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

**Evaluate Rational Expressions**

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

$\frac{4p-1}{{p}^{2}+5}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=\mathrm{-1}$

$\frac{{q}^{2}-5}{q+3}\phantom{\rule{0.2em}{0ex}}\text{when q}=7$

$\frac{{y}^{2}-8}{{y}^{2}-y-2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=1$

$\frac{{z}^{2}+2}{4z-{z}^{2}}\phantom{\rule{0.2em}{0ex}}\text{when z}=3$

**Simplify Rational Expressions**

In the following exercises, simplify.

$\frac{8{m}^{4}}{16m{n}^{3}}$

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

**Simplify Rational Expressions with Opposite Factors**

In the following exercises, simplify.

$\frac{d-16}{16-d}$

$\frac{{w}^{2}-3w-28}{49-{w}^{2}}$

##### Multiply and Divide Rational Expressions

**Multiply Rational Expressions**

In the following exercises, multiply.

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

$\frac{5{z}^{2}}{5{z}^{2}+40z+35}\xb7\frac{{z}^{2}-1}{3z}$

**Divide Rational Expressions**

In the following exercises, divide.

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

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

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

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

**Add Rational Expressions with a Common Denominator**

In the following exercises, add.

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

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

**Subtract Rational Expressions with a Common Denominator**

In the following exercises, subtract.

$\frac{{z}^{2}}{z+10}-\frac{100}{z+10}$

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

**Add and Subtract Rational Expressions whose Denominators are Opposites**

In the following exercises, add and subtract.

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

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

##### Add and Subtract Rational Expressions With Unlike Denominators

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

In the following exercises, find the LCD.

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

**Find Equivalent Rational Expressions**

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

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

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

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

**Add Rational Expressions with Different Denominators**

In the following exercises, add.

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

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

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

**Subtract Rational Expressions with Different Denominators**

In the following exercises, subtract and add.

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

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

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

##### 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.

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

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

##### 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 a Rational Equation for a Specific Variable**

In the following exercises, solve for the indicated variable.

$\frac{1}{x}-\frac{2}{y}=5\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}y$

$P=\frac{k}{V}\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}V$

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

**Solve Proportions**

In the following exercises, solve.

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

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

In the following exercises, solve using proportions.

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.

∆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.

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

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.

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.

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?

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?

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?

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.

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

Lisa 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?

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.

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.

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?