Chapter 7 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.

##

Worked example

complex rational expression

A complex rational expression is a rational expression in which the numerator and/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

When two rational expressions are equal, the equation relating them is called a proportion.

rational equation

A rational equation is an equation that contains a rational expression.

rational expression

p

q

rational function

rational inequality

A rational inequality is an inequality that contains a rational expression.

similar figures

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

simplified rational expression

A simplified rational expression has no common factors, other than 1, in its numerator and denominator.

zero partition number of a rational inequality

The zero partition number of a rational inequality is a number which makes the rational expression zero or undefined.

##

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.

• Equivalent Fractions Property If a, b, and c are numbers where $b\ne{}0,c\ne{}0,$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• How to 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.$$\frac{a-b}{b-a}=-1\,a\ne{}b$ An expression and its opposite divide to $-1.$

• Multiplication of Rational Expressions If p, q, r, and s are polynomials where $q\ne{}0,s\ne{}0,$ then $\,\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}$

• How to multiply rational expressions. 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, and s are polynomials where $q\ne{}0,r\ne{}0,s\ne{}0,$ then $\,\frac{p}{q}\mathrm{÷}\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}$

• How to 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.

• How to determine the domain of a rational function. Step 1. Set the denominator equal to zero. Step 2. Solve the equation. Step 3. The domain is all real numbers excluding the values found in Step 2.

• Rational Expression Addition and Subtraction If p, q, and r are polynomials where $r\ne{}0,$ then $\,\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$ and $\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}$

• How to 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. Write the LCD as the product of the factors.

• How to 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.

• How to simplify a complex 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.

• How 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.

• How 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.

• A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d},$ where $b\ne{}0,d\ne{}0.$ The proportion is read “a is to b as c is to d.”

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

• Direct Variation For any two variables x and y, y varies directly with x if $y=kx,$ where $k\ne{}0.$ The constant k is called the constant of variation. How to solve direct variation problems. Step 1. Write the formula for direct variation. Step 2. Substitute the given values for the variables. Step 3. Solve for the constant of variation. Step 4. Write the equation that relates $x$ and $y.$

• Inverse Variation For any two variables x and y, y varies inversely with x if $y=\frac{k}{x},$ where $k\ne{}0.$ The constant k is called the constant of variation. How to solve inverse variation problems. Step 1. Write the formula for inverse variation. Step 2. Substitute the given values for the variables. Step 3. Solve for the constant of variation. Step 4. Write the equation that relates x and y.

• Solve a rational inequality. Step 1. Write the inequality as one quotient on the left and zero on the right. Step 2. Determine the zero partition numbers–the points where the rational expression will be zero or undefined. Step 3. Use the zero partition numbers to divide the number line into intervals. Step 4. Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

##

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.

Simplify Rational Expressions

In the following exercises, simplify.

Multiply Rational Expressions

In the following exercises, multiply.

Divide Rational Expressions

In the following exercises, divide.

Multiply and Divide Rational Functions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

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.

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

Add and Subtract Rational Functions

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

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

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 Rational Equations that Involve Functions

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Solve Proportions

In the following exercises, solve.

Solve Using Proportions

In the following exercises, solve.

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.

Solve Rational Inequalities

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

Solve an Inequality with Rational Functions

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

##

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 each rational inequality and write the solution in interval notation.

In the following exercises, find $R(x)$ given $f(x)=\frac{x-4}{{x}^{2}-3x-10}$ and $g(x)=\frac{x-5}{{x}^{2}-2x-8}.$

In the following exercises, solve.

Key Terms

complex rational expression

A complex rational expression is a rational expression in which the numerator and/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

When two rational expressions are equal, the equation relating them is called a proportion.

rational equation

A rational equation is an equation that contains a rational expression.

rational expression

p

q

rational function

rational inequality

A rational inequality is an inequality that contains a rational expression.

similar figures

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

simplified rational expression

A simplified rational expression has no common factors, other than 1, in its numerator and denominator.

zero partition number of a rational inequality

The zero partition number of a rational inequality is a number which makes the rational expression zero or undefined.

##

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.

• Equivalent Fractions Property If a, b, and c are numbers where $b\ne{}0,c\ne{}0,$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• How to 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.$$\frac{a-b}{b-a}=-1\,a\ne{}b$ An expression and its opposite divide to $-1.$

• Multiplication of Rational Expressions If p, q, r, and s are polynomials where $q\ne{}0,s\ne{}0,$ then $\,\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}$

• How to multiply rational expressions. 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, and s are polynomials where $q\ne{}0,r\ne{}0,s\ne{}0,$ then $\,\frac{p}{q}\mathrm{÷}\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}$

• How to 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.

• How to determine the domain of a rational function. Step 1. Set the denominator equal to zero. Step 2. Solve the equation. Step 3. The domain is all real numbers excluding the values found in Step 2.

• Rational Expression Addition and Subtraction If p, q, and r are polynomials where $r\ne{}0,$ then $\,\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$ and $\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}$

• How to 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. Write the LCD as the product of the factors.

• How to 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.

• How to simplify a complex 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.

• How 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.

• How 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.

• A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d},$ where $b\ne{}0,d\ne{}0.$ The proportion is read “a is to b as c is to d.”

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

• Direct Variation For any two variables x and y, y varies directly with x if $y=kx,$ where $k\ne{}0.$ The constant k is called the constant of variation. How to solve direct variation problems. Step 1. Write the formula for direct variation. Step 2. Substitute the given values for the variables. Step 3. Solve for the constant of variation. Step 4. Write the equation that relates $x$ and $y.$

• Inverse Variation For any two variables x and y, y varies inversely with x if $y=\frac{k}{x},$ where $k\ne{}0.$ The constant k is called the constant of variation. How to solve inverse variation problems. Step 1. Write the formula for inverse variation. Step 2. Substitute the given values for the variables. Step 3. Solve for the constant of variation. Step 4. Write the equation that relates x and y.

• Solve a rational inequality. Step 1. Write the inequality as one quotient on the left and zero on the right. Step 2. Determine the zero partition numbers–the points where the rational expression will be zero or undefined. Step 3. Use the zero partition numbers to divide the number line into intervals. Step 4. Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

##

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.

Simplify Rational Expressions

In the following exercises, simplify.

Multiply Rational Expressions

In the following exercises, multiply.

Divide Rational Expressions

In the following exercises, divide.

Multiply and Divide Rational Functions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

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.

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

Add and Subtract Rational Functions

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

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

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 Rational Equations that Involve Functions

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Solve Proportions

In the following exercises, solve.

Solve Using Proportions

In the following exercises, solve.

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.

Solve Rational Inequalities

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

Solve an Inequality with Rational Functions

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

##

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 each rational inequality and write the solution in interval notation.

In the following exercises, find $R(x)$ given $f(x)=\frac{x-4}{{x}^{2}-3x-10}$ and $g(x)=\frac{x-5}{{x}^{2}-2x-8}.$

In the following exercises, solve.

1. $\frac{5a+3}{3a-2}$

2. $\frac{5{x}^{2}{y}^{2}}{8y}$

3. $\frac{18}{24}$

4. $\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}$

5. $\frac{5}{8}\cdot\frac{4}{15}$

1. $\frac{72x-12{x}^{2}}{8x+32}\cdot\frac{{x}^{2}+10x+24}{{x}^{2}-36}$

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

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

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

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

1. $\frac{7}{15}+\frac{8}{15}$

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

3. $\frac{{x}^{2}}{x-7}-\frac{3x+28}{x-7}$

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

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

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

2. $\frac{7}{{a}^{2}-3a-10},\frac{3a}{{a}^{2}-a-20}$

3. $\frac{5}{3{p}^{2}+17p-6},\frac{2m}{3{p}^{2}+23p-8}$

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

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

1. $\frac{3y}{y+2}-\frac{y+2}{y+8}$

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

3. $\frac{8a}{{a}^{2}-64}-\frac{4}{a+8}$

4. $f(x)=\frac{2{x}^{2}+12x-11}{{x}^{2}+3x-10},$$g(x)=\frac{x+1}{2-x}$

5. $f(x)=\frac{4x}{{x}^{2}-121},$$g(x)=\frac{2}{x-11}$

1. $\frac{\frac{7x}{x+2}}{\frac{14{x}^{2}}{{x}^{2}-4}}$

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

3. $\frac{\frac{1}{3}+\frac{1}{8}}{\frac{1}{4}+\frac{1}{12}}$

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

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

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

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

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

4. $\frac{V}{l}=hw$ for $l.$

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

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

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

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

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

5. 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.

1. 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?

2. 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?

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. $\frac{x-3}{x+4}\le{}0$

5. $\frac{3x-2}{x-4}\le{}2$

1. $\frac{1}{2}-\frac{4}{{x}^{2}}\ge\frac{1}{x}$

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

3. The function$C(x)=150x+100,000$ 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.

4. $\frac{4{a}^{2}b}{12a{b}^{2}}$

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

1. $\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}$

2. $\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}$

3. $\frac{\frac{1}{m}-\frac{1}{n}}{\frac{1}{n}+\frac{1}{m}}$

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

5. $\frac{6x}{x-6}\le{}2$

1. $\frac{1}{2}+\frac{12}{{x}^{2}}\ge\frac{5}{x}$

2. $R(x)=f(x)\cdot{}g(x)$

3. Given the function,$R(x)=\frac{2}{2{x}^{2}+x-15},$ find the values of $x$ that make the function less than or equal to 0.

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

5. Oliver 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 Oliver’s dad working alone to split the logs?

1. 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?

2. 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?

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

4. 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?

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

1. The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The diagram 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?