Chapter 8 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

complex rational expression

A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression.

extraneous solution to a rational equation

An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.

proportion

rational equation

A rational equation is two rational expressions connected by an equal sign.

rational expression

p

q

similar figures

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

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Key Concepts

• Determine the Values for Which a Rational Expression is Undefined Step 1. Set the denominator equal to zero. Step 2. Solve the equation, if possible.

• Simplified Rational Expression A rational expression is considered simplified if there are no common factors in its numerator and denominator.

• Simplify a Rational Expression Step 1. Factor the numerator and denominator completely. Step 2. Simplify by dividing out common factors.

• Opposites in a Rational Expression The opposite of $a-b$ is $b-a$.$\begin{array}{lccccl} \frac{a-b}{b-a}=-1 & & & & & a\ne{}0,b\ne{}0,a\ne{}b \\ \end{array}$

• Multiplication of Rational Expressions If $p,q,r,s$ are polynomials where $q\ne{}0,s\ne{}0$, then $\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}$. To multiply rational expressions, multiply the numerators and multiply the denominators

• Multiply a Rational Expression Step 1. Factor each numerator and denominator completely. Step 2. Multiply the numerators and denominators. Step 3. Simplify by dividing out common factors.

• Division of Rational Expressions If $p,q,r,s$ are polynomials where $q\ne{}0,r\ne{}0,s\ne{}0$, then $\frac{p}{q}\mathrm{÷}\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}$. To divide rational expressions multiply the first fraction by the reciprocal of the second.

• Divide Rational Expressions Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second. Step 2. Factor the numerators and denominators completely. Step 3. Multiply the numerators and denominators together. Step 4. Simplify by dividing out common factors.

• Rational Expression Addition If $p,q,\,\mathrm{and}\,r$ are polynomials where $r\ne{}0$, then

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

• Rational Expression Subtraction If $p,q,\,\mathrm{and}\,r$ are polynomials where $r\ne{}0$, then

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

• Find the Least Common Denominator of Rational Expressions Step 1. Factor each expression completely. Step 2. List the factors of each expression. Match factors vertically when possible. Step 3. Bring down the columns. Step 4. Multiply the factors.

• Add or Subtract Rational Expressions Step 1. Determine if the expressions have a common denominator. Yes – go to step 2. No – Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. Step 2. Add or subtract the rational expressions. Step 3. Simplify, if possible.

• To Simplify a Rational Expression by Writing it as Division Step 1. Simplify the numerator and denominator. Step 2. Rewrite the complex rational expression as a division problem. Step 3. Divide the expressions.

• To Simplify a Complex Rational Expression by Using the LCD Step 1. Find the LCD of all fractions in the complex rational expression. Step 2. Multiply the numerator and denominator by the LCD. Step 3. Simplify the expression.

• Strategy to Solve Equations with Rational Expressions Step 1. Note any value of the variable that would make any denominator zero. Step 2. Find the least common denominator of all denominators in the equation. Step 3. Clear the fractions by multiplying both sides of the equation by the LCD. Step 4. Solve the resulting equation. Step 5. Check. If any values found in Step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation.

• Property of Similar Triangles If $\Delta{}ABC$ is similar to $\Delta{}XYZ$, then their corresponding angle measures are equal and their corresponding sides are in the same ratio.

• Problem Solving Strategy for Geometry Applications Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

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Review Exercises

Determine the Values for Which a Rational Expression is Undefined

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

Evaluate Rational Expressions

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

Simplify Rational Expressions

In the following exercises, simplify.

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

Multiply Rational Expressions

In the following exercises, multiply.

Divide Rational Expressions

In the following exercises, divide.

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

Find Equivalent Rational Expressions

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

Add Rational Expressions with Different Denominators

In the following exercises, add.

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Solve Rational Equations

In the following exercises, solve.

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Solve Proportions

In the following exercises, solve.

In the following exercises, solve using proportions.

Solve Similar Figure Applications

In the following exercises, solve.

Solve Uniform Motion Applications

In the following exercises, solve.

Solve Work Applications

In the following exercises, solve.

Solve Direct Variation Problems

In the following exercises, solve.

Solve Inverse Variation Problems

In the following exercises, solve.

##

In the following exercises, simplify.

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

In the following exercises, solve each equation.

In the following exercises, solve.

Key Terms

complex rational expression

A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression.

extraneous solution to a rational equation

An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.

proportion

rational equation

A rational equation is two rational expressions connected by an equal sign.

rational expression

p

q

similar figures

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

##

Key Concepts

• Determine the Values for Which a Rational Expression is Undefined Step 1. Set the denominator equal to zero. Step 2. Solve the equation, if possible.

• Simplified Rational Expression A rational expression is considered simplified if there are no common factors in its numerator and denominator.

• Simplify a Rational Expression Step 1. Factor the numerator and denominator completely. Step 2. Simplify by dividing out common factors.

• Opposites in a Rational Expression The opposite of $a-b$ is $b-a$.$\begin{array}{lccccl} \frac{a-b}{b-a}=-1 & & & & & a\ne{}0,b\ne{}0,a\ne{}b \\ \end{array}$

• Multiplication of Rational Expressions If $p,q,r,s$ are polynomials where $q\ne{}0,s\ne{}0$, then $\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}$. To multiply rational expressions, multiply the numerators and multiply the denominators

• Multiply a Rational Expression Step 1. Factor each numerator and denominator completely. Step 2. Multiply the numerators and denominators. Step 3. Simplify by dividing out common factors.

• Division of Rational Expressions If $p,q,r,s$ are polynomials where $q\ne{}0,r\ne{}0,s\ne{}0$, then $\frac{p}{q}\mathrm{÷}\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}$. To divide rational expressions multiply the first fraction by the reciprocal of the second.

• Divide Rational Expressions Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second. Step 2. Factor the numerators and denominators completely. Step 3. Multiply the numerators and denominators together. Step 4. Simplify by dividing out common factors.

• Rational Expression Addition If $p,q,\,\mathrm{and}\,r$ are polynomials where $r\ne{}0$, then

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

• Rational Expression Subtraction If $p,q,\,\mathrm{and}\,r$ are polynomials where $r\ne{}0$, then

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

• Find the Least Common Denominator of Rational Expressions Step 1. Factor each expression completely. Step 2. List the factors of each expression. Match factors vertically when possible. Step 3. Bring down the columns. Step 4. Multiply the factors.

• Add or Subtract Rational Expressions Step 1. Determine if the expressions have a common denominator. Yes – go to step 2. No – Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. Step 2. Add or subtract the rational expressions. Step 3. Simplify, if possible.

• To Simplify a Rational Expression by Writing it as Division Step 1. Simplify the numerator and denominator. Step 2. Rewrite the complex rational expression as a division problem. Step 3. Divide the expressions.

• To Simplify a Complex Rational Expression by Using the LCD Step 1. Find the LCD of all fractions in the complex rational expression. Step 2. Multiply the numerator and denominator by the LCD. Step 3. Simplify the expression.

• Strategy to Solve Equations with Rational Expressions Step 1. Note any value of the variable that would make any denominator zero. Step 2. Find the least common denominator of all denominators in the equation. Step 3. Clear the fractions by multiplying both sides of the equation by the LCD. Step 4. Solve the resulting equation. Step 5. Check. If any values found in Step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation.

• Property of Similar Triangles If $\Delta{}ABC$ is similar to $\Delta{}XYZ$, then their corresponding angle measures are equal and their corresponding sides are in the same ratio.

• Problem Solving Strategy for Geometry Applications Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

##

Review Exercises

Determine the Values for Which a Rational Expression is Undefined

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

Evaluate Rational Expressions

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

Simplify Rational Expressions

In the following exercises, simplify.

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

Multiply Rational Expressions

In the following exercises, multiply.

Divide Rational Expressions

In the following exercises, divide.

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

Find Equivalent Rational Expressions

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

Add Rational Expressions with Different Denominators

In the following exercises, add.

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Solve Rational Equations

In the following exercises, solve.

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Solve Proportions

In the following exercises, solve.

In the following exercises, solve using proportions.

Solve Similar Figure Applications

In the following exercises, solve.

Solve Uniform Motion Applications

In the following exercises, solve.

Solve Work Applications

In the following exercises, solve.

Solve Direct Variation Problems

In the following exercises, solve.

Solve Inverse Variation Problems

In the following exercises, solve.

##

In the following exercises, simplify.

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

In the following exercises, solve each equation.

In the following exercises, solve.

1. $\frac{2a+1}{3a-2}$

2. $\frac{3x{y}^{2}}{5y}$

3. $\frac{4p-1}{{p}^{2}+5}\,\mathrm{when}\,p=-1$

4. $\frac{{y}^{2}-8}{{y}^{2}-y-2}\,\mathrm{when}\,y=1$

1. $\frac{10}{24}$

2. $\frac{14a-14}{a-1}$

3. $\frac{{c}^{2}-c-2}{4-{c}^{2}}$

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

1. $\frac{3}{8}\cdot\frac{2}{15}$

2. $\frac{3{a}^{2}+21a}{{a}^{2}+6a-7}\cdot\frac{a-1}{ab}$

3. $\frac{{t}^{2}-4t-12}{{t}^{2}+8t+12}\mathrm{÷}\frac{{t}^{2}-36}{6t}$

4. $\frac{11+w}{w-9}\mathrm{÷}\frac{121-{w}^{2}}{9-w}$

1. $\frac{\frac{{c}^{2}-64}{3{c}^{2}+26c+16}}{\frac{{c}^{2}-4c-32}{15c+10}}$

2. $\frac{3}{5}+\frac{2}{5}$

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

4. $\frac{{p}^{2}+10p}{p+5}+\frac{25}{p+5}$

1. $\frac{{d}^{2}}{d+4}-\frac{3d+28}{d+4}$

2. $\frac{4{q}^{2}-q+3}{{q}^{2}+6q+5}-\frac{3{q}^{2}+q+6}{{q}^{2}+6q+5}$

3. $\frac{18w}{6w-1}+\frac{3w-2}{1-6w}$

4. $\frac{2{b}^{2}+3b-15}{{b}^{2}-49}-\frac{{b}^{2}+16b-1}{49-{b}^{2}}$

1. $\frac{4}{{m}^{2}-3m-10},\frac{2m}{{m}^{2}-m-20}$

2. $\frac{5}{3{p}^{2}+19p+6},\frac{2p}{3{p}^{2}+25p+8}$

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

4. $\frac{2}{3}+\frac{3}{5}$

1. $\frac{2}{c-2}+\frac{9}{c+3}$

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

3. $\frac{3v}{v+2}-\frac{v+2}{v+8}$

4. $\frac{7m+3}{m+2}-5$

1. $\frac{8d}{{d}^{2}-64}-\frac{4}{d+8}$

2. $\frac{\frac{5a}{a+2}}{\frac{10{a}^{2}}{{a}^{2}-4}}$

3. $\frac{x-\frac{3x}{x+5}}{\frac{1}{x+5}+\frac{1}{x-5}}$

4. $\frac{6+\frac{2}{q-4}}{\frac{5}{q+4}}$

1. $\frac{\frac{2}{{z}^{2}-49}+\frac{1}{z+7}}{\frac{9}{z+7}+\frac{12}{z-7}}$

2. $\frac{1}{2}+\frac{2}{3}=\frac{1}{x}$

3. $\frac{1}{b-2}+\frac{1}{b+2}=\frac{3}{{b}^{2}-4}$

4. $\frac{v-15}{{v}^{2}-9v+18}=\frac{4}{v-3}+\frac{2}{v-6}$

1. $\frac{V}{l}=hw\,\mathrm{for}\,l$

2. $x=\frac{y+5}{z-7}\,\mathrm{for}\,z$

3. $\frac{x}{4}=\frac{3}{5}$

4. $\frac{s}{s+20}=\frac{3}{7}$

1. 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?

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

3. 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?

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

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

2. 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?

3. 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 each, how many tickets can Madelyn afford to buy if the price increased to$8?

4. $\frac{3{a}^{2}b}{6a{b}^{2}}$

1. $\frac{4x}{x+2}\cdot\frac{{x}^{2}+5x+6}{12{x}^{2}}$

2. $\frac{4}{pq}+\frac{5}{p}$

3. $\frac{\frac{2}{3}+\frac{3}{5}}{\frac{2}{5}}$

4. $\frac{1}{2}+\frac{2}{7}=\frac{1}{x}$

1. $\frac{1}{z-5}+\frac{1}{z+5}=\frac{1}{{z}^{2}-25}$

2. 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?

3. 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?

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

2. ∆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.

3. 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.

4. 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?

1. 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?

2. 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?