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Section P.5 Rational Expressions

Before Starting This Section, Review

1 Rational numbers (Section P.1 , page 3)
2 Properties of fractions (Section P.1 , page 13)
3 Factoring polynomials (Section P.4 )
4 Irreducible polynomial (Section P.4 , page 42)

Objectives

1 Define rational expressions.
2 Reduce a rational expression to lowest terms.
3 Multiply and divide rational expressions.
4 Add and subtract rational expressions.
5 Identify and simplify complex fractions.

False Positive in Drug Testing

You are probably aware that the nonmedical use of steroids among adolescents and young adults is an ongoing national concern. In fact, many high schools have required students involved in extracurricular activities to submit to random drug testing. Many methods of drug testing are in use, and those who administer the tests are keenly aware of the possibility of what is termed a “false positive,” which occurs when someone who is not a drug user tests positive for using drugs. How likely is it that a student who tests positive actually uses drugs? For a test that is 95% accurate, an expression for the likelihood that a student who tests positive but is not a drug user is

$\frac{(0.05) (1 - x)}{0.95 x + (0.05) (1 - x)},$

where x is the percent of the student population that uses drugs. This is an example of a rational expression, which we study in this section. In Example 1, we see that in a student population among which only 5% of the students use drugs, the likelihood that a student who tests positive is not a drug user is 50% when the test used is 95% accurate.

Rational Expressions

1 Define rational expressions.

Recall that the quotient of two integers, $\frac{a}{b} (b \neq 0),$ is a rational number. The quotient of two polynomials is called a rational expression. Following are examples of rational expressions.

$\frac{10}{17}, \frac{x + 2}{3}, \frac{x + 6}{(x - 3) (x + 4)}, \frac{7}{x^{2} + 1}, and \frac{x^{2} + 2 x - 3}{x^{2} + 5 x}$

We use the same language to describe rational expressions that we use to describe rational numbers. For $\frac{x^{2} + 2 x - 3}{x^{2} + 5 x + 6}, {we call x}^{2} + 2 x - 3$ the numerator and $x^{2} + 5 x + 6$ the denominator. The domain of a rational expression is the set of all real numbers except those that result in a zero denominator. For example, the domain of $\frac{2}{x - 1}$ is all real $x, x \neq 1 .$ We write $\frac{x + 6}{(x - 3) (x + 4)}, x \neq 3, x \neq - 4$ to indicate that 3 and $- 4$ are not in the domain of this rational expression.

Example 1 Calculating False Positives in Drug Testing

With a test that is 95% accurate, an expression for the likelihood that a student who tests positive for drugs is a nonuser is

$\frac{(0.05) (1 - x)}{0.95 x + (0.05) (1 - x)},$

where x is the percent (in decimal) of the student population that uses drugs. In a student population among which only 5% of the students use drugs, find the likelihood that a student who tests positive is a nonuser.

Solution

We convert 5% to a decimal to get $x = 0.05,$ and we replace x with 0.05 in the given expression.

$\begin{array}{rcll} \frac{(0.05) (1 - x)}{0.95 x + (0.05) (1 - x)} & = & \frac{(0.05) (1 - 0.05)}{0.95 (0.05) + (0.05) (1 - 0.05)} \\ = & 0.50 & Use a calculator . \end{array}$

So the likelihood that a student who tests positive is a nonuser when the test used is 95% accurate is 50%. Because of the high rate of false positives, a more accurate test is needed.

Practice Problem 1

Rework Example 1 for a student population among which 10% of the students use drugs.

Lowest Terms for a Rational Expression

2 Reduce a rational expression to lowest terms.

A rational expression is reduced to its lowest terms or simplified if its numerator and denominator have no common factor other than 1 or $- 1 .$

Side Note

The following property of fractions (see page 13)

$\frac{a c}{b c} = \frac{a}{b}, b \neq 0, c \neq 0$

is used to simplify rational expressions.

Example 2 Reducing a Rational Expression to Lowest Terms

Simplify each expression.

$\frac{x^{4} + 2 x^{3}}{x + 2}, x \neq - 2$
$\frac{x^{2} - x - 6}{x^{3} - 3 x^{2}}, x \neq 0, x \neq 3$

Solution

Factor each numerator and denominator and divide out the common factors.

$\frac{x^{4} + 2 x^{3}}{x + 2} = \frac{x^{3} (x + 2)}{x + 2} = \frac{x^{3} (x + 2)}{(x + 2)} = x^{3}$
$\frac{x^{2} - x - 6}{x^{3} - 3 x^{2}} = \frac{(x + 2) (x - 3)}{x^{2} (x - 3)} = \frac{(x + 2) (x - 3)}{x^{2} (x - 3)} = \frac{x + 2}{x^{2}}$

Practice Problem 2

Simplify each expression.
1. $\frac{2 x^{3} + 8 x^{2}}{3 x^{2} + 12 x}$
2. $\frac{x^{2} - 4}{x^{2} + 4 x + 4}$

Warning

Because $x^{2}$ is a term, not a factor, in both the numerator and denominator of $\frac{x^{2} - 1}{x^{2} + 2 x + 1},$ you cannot remove $x^{2} .$ You can remove only factors common to both the numerator and denominator of a rational expression.

Multiplication and Division of Rational Expressions

3 Multiply and divide rational expressions.

We use the same rules for multiplying and dividing rational expressions as we do for rational numbers.

Side Note

In working with the quotient of two rational expressions $\frac{\frac{A}{B}}{\frac{C}{D}} = \frac{A}{B} \cdot \frac{D}{C}$ three polynomials appear as denominators: B and D from $\frac{A}{B}$ and $\frac{C}{D}$ and C from $\frac{A}{B} \cdot \frac{D}{C} .$

Side Note

The best strategy to use when multiplying and dividing rational expressions is to factor each numerator and denominator completely and then divide out the common factors.

Example 3 Multiplying and Dividing Rational Expressions

Multiply or divide as indicated. Simplify your answer and leave it in factored form.

$\frac{x^{2} + 3 x + 2}{x^{3} + 3 x} \cdot \frac{2 x^{3} + 6 x}{x^{2} + x - 2},$ $x \neq 0, x \neq 1, x \neq - 2$
$\frac{\frac{3 x^{2} + 11 x - 4}{8 x^{3} - 40 x^{2}}}{\frac{3 x + 12}{4 x^{4} - 20 x^{3}}}, x \neq 0, x \neq 5, x \neq - 4$

Solution

Factor each numerator and denominator and divide out the common factors.

$\begin{array}{rcl} \frac{x^{2} + 3 x + 2}{x^{3} + 3 x} \cdot \frac{2 x^{3} + 6 x}{x^{2} + x - 2} & = & \frac{(x + 2) (x + 1)}{x (x^{2} + 3)} \cdot \frac{2 x (x^{2} + 3)}{(x - 1) (x + 2))} \\ = & \frac{(x + 2) (x + 1) (2 x) (x^{2} + 3)}{x (x^{2} + 3) (x - 1) (x + 2)} = \frac{2 (x + 1)}{x - 1} \end{array}$
$\begin{array}{rcl} \frac{\frac{3 x^{2} + 11 x - 4}{8 x^{3} - 40 x^{2}}}{\frac{3 x + 12}{4 x^{4} - 20 x^{3}}} & = & \frac{3 x^{2} + 11 x - 4}{8 x^{3} - 40 x^{2}} \cdot \frac{4 x^{4} - 20 x^{3}}{3 x + 12} = \frac{(x + 4) (3 x - 1)}{8 x^{2} (x - 5)} \cdot \frac{4 x^{3} (x - 5)}{3 (x + 4)} \\ = & \frac{(x + 4) (3 x - 1) \overset{x}{(4 x^{3})} (x - 5)}{\underset{2}{8 x^{2}} (x - 5) (3) (x + 4)} = \frac{(3 x - 1) x}{6} = \frac{x (3 x - 1)}{6} \end{array}$

Practice Problem 3

Multiply or divide as indicated. Simplify your answer.

$\frac{\frac{x^{2} - 2 x - 3}{7 x^{3} + 28 x^{2}}}{\frac{4 x + 4}{2 x^{4} + 8 x^{3}}} x \neq 0, x \neq - 4, x \neq - 1$

Addition and Subtraction of Rational Expressions

4 Add and subtract rational expressions.

To add and subtract rational expressions, we use the same rules as for adding and subtracting rational numbers.

Example 4 Adding and Subtracting Rational Expressions with the Same Denominator

Add or subtract as indicated. Simplify your answer and leave both the numerator and denominator in factored form.

$\frac{x - 6}{{(x + 1)}^{2}} + \frac{x + 8}{{(x + 1)}^{2}}, x \neq - 1$
$\frac{3 x - 2}{x^{2} - 5 x + 6} - \frac{2 x + 1}{x^{2} - 5 x + 6}, x \neq 2, x \neq 3$

Solution

$\begin{array}{rcl} \frac{x - 6}{{(x + 1)}^{2}} + \frac{x + 8}{{(x + 1)}^{2}} & = & \frac{x - 6 + x + 8}{{(x + 1)}^{2}} = \frac{2 x + 2}{{(x + 1)}^{2}} \\ = & \frac{2 (x + 1)}{(x + 1) (x + 1)} = \frac{2}{x + 1} \end{array}$
$\begin{array}{rcl} \frac{3 x - 2}{x^{2} - 5 x + 6} - \frac{2 x + 1}{x^{2} - 5 x + 6} & = & \frac{(3 x - 2) - (2 x + 1)}{x^{2} - 5 x + 6} = \frac{3 x - 2 - 2 x - 1}{x^{2} - 5 x + 6} \\ = & \frac{x - 3}{(x - 2) (x - 3)} = \frac{x - 3}{(x - 2) (x - 3)} = \frac{1}{x - 2} \end{array}$

Practice Problem 4

Add or subtract as indicated. Simplify your answers.
1. $\frac{5 x + 22}{x^{2} - 36} + \frac{2 (x + 10)}{x^{2} - 36}, x \neq 6, x \neq - 6$
2. $\frac{4 x + 1}{x^{2} + x - 12} - \frac{3 x + 4}{x^{2} + x - 12}, x \neq - 4, x \neq 3$

When adding or subtracting rational expressions with different denominators, we proceed (as with fractions) by finding a common denominator. The one most convenient to use is called the least common denominator (LCD), and it is the polynomial of least degree that contains each denominator as a factor. In the simplest case, the LCD is the product of the denominators.

Example 5 Adding and Subtracting When Denominators Have No Common Factor

Add or subtract as indicated. Simplify your answer and leave it in factored form.

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

Solution

$\begin{array}{rcll} \frac{x}{x + 1} + \frac{2 x - 1}{x + 3} & = & \frac{x (x + 3)}{(x + 1) (x + 3)} + \frac{(2 x - 1) (x + 1)}{(x + 1) (x + 3)} & \begin{array}{l} LCD = \\ (x + 1) (x + 3) \end{array} \\ = & \frac{x (x + 3) + (2 x - 1) (x + 1)}{(x + 1) (x + 3)} \\ = & \frac{x^{2} + 3 x + 2 x^{2} + 2 x - x - 1}{(x + 1) (x + 3)} \\ = & \frac{3 x^{2} + 4 x - 1}{(x + 1) (x + 3)} \end{array}$
$\begin{array}{rcl} \frac{2 x}{x + 1} - \frac{x}{x + 2} & = & \frac{2 x (x + 2)}{(x + 1) (x + 2)} - \frac{x (x + 1)}{(x + 1) (x + 2)} \begin{array}{l} LCD = \\ (x + 1) (x + 2) \end{array} \\ = & \frac{2 x (x + 2) - x (x + 1)}{(x + 1) (x + 2)} = \frac{2 x^{2} + 4 x - x^{2} - x}{(x + 1) (x + 2)} \\ = & \frac{x^{2} + 3 x}{(x + 1) (x + 2)} = \frac{x (x + 3)}{(x + 1) (x + 2)} \end{array}$

Practice Problem 5

Add or subtract as indicated. Simplify your answers.
1. $\frac{2 x}{x + 2} + \frac{3 x}{x - 5}, x \neq - 2, x \neq 5$
2. $\frac{5 x}{x - 4} - \frac{2 x}{x + 3}, x \neq 4, x \neq - 3$

Example 6 Finding the LCD for Rational Expressions

Find the LCD for each pair of rational expressions.

$\frac{x + 2}{x {(x - 1)}^{2} (x + 2)}, \frac{3 x + 7}{4 x^{2} {(x + 2)}^{3}}$
$\frac{x + 1}{x^{2} - x - 6}, \frac{2 x - 13}{x^{2} - 9}$

Solution

1. Step 1 The denominators are already completely factored.
2. Step 2 $4 x (x - 1) (x + 2)$ Product of the different factors
3. Step 3 $LCD = 4 x^{2} {(x - 1)}^{2} {(x + 2)}^{3}$ The greatest exponents are 2, 2, and 3.
1. Step 1 $\begin{array}{rcl} x^{2} - x - 6 & = & (x + 2) (x - 3) \\ x^{2} - 9 & = & (x + 3) (x - 3) \end{array}$
2. Step 2 $(x + 2) (x - 3) (x + 3)$ Product of the different factors
3. Step 3 $LCD = (x + 2) (x - 3) (x + 3)$ The greatest exponent on each factor is 1.

Practice Problem 6

Find the LCD for each pair of rational expressions.
1. $\frac{x^{2} + 3 x}{x^{2} {(x + 2)}^{2} (x - 2)}, \frac{4 x^{2} + 1}{3 x {(x - 2)}^{2}}$
2. $\frac{2 x - 1}{x^{2} - 25}, \frac{3 - 7 x^{2}}{(x^{2} + 4 x - 5)}$

When adding or subtracting rational expressions with different denominators, the first step is to find the LCD. Here is the general procedure.

Example 7 Using the LCD to Add and Subtract Rational Expressions

Add or subtract as indicated. Simplify your answer and leave it in factored form.

$\frac{3}{x^{2} - 1} + \frac{x}{x^{2} + 2 x + 1}, x \neq 1, x \neq - 1$
$\frac{x + 2}{x^{2} - x} - \frac{3 x}{4 {(x - 1)}^{2}}, x \neq 0, x \neq 1$

Solution

1. Step 1 Note that $x^{2} - 1 = (x - 1) (x + 1) {and x}^{2} + 2 x + 1 = {(x + 1)}^{2};$ the LCD is ${(x + 1)}^{2} (x - 1) .$
2. Step 2
  
  $\begin{array}{l} \frac{3}{x^{2} - 1} = \frac{3}{(x - 1) (x + 1)} = \frac{3 (x + 1)}{(x - 1) {(x + 1)}^{2}} & \begin{array}{l} Multiply numerator \\ and denominator by \\ x + 1. \end{array} \\ \frac{x}{x^{2} + 2 x + 1} = \frac{x}{{(x + 1)}^{2}} = \frac{x (x - 1)}{{(x + 1)}^{2} (x - 1)} & \begin{array}{l} Multiply numerator \\ and denominator by \\ x - 1. \end{array} \end{array}$
3. Steps 3–4 Add.
  
  $\begin{array}{rcl} \frac{3}{x^{2} - 1} + \frac{x}{x^{2} + 2 x + 1} & = & \frac{3 (x + 1)}{(x - 1) {(x + 1)}^{2}} + \frac{x (x - 1)}{(x - 1) {(x + 1)}^{2}} \\ = & \frac{3 (x + 1) + x (x - 1)}{(x - 1) {(x + 1)}^{2}} = \frac{3 x + 3 + x^{2} - x}{(x - 1) {(x + 1)}^{2}} \\ = & \frac{x^{2} + 2 x + 3}{(x - 1) {(x + 1)}^{2}} \end{array}$
1. Step 1 The LCD is $4 x {(x - 1)}^{2}$ . $x^{2} - x = x (x - 1)$
2. Step 2
  
  $\begin{array}{l} \frac{x + 2}{x^{2} - x} = \frac{x + 2}{x (x - 1)} = \frac{(x + 2) \cdot 4 (x - 1)}{x (x - 1) \cdot 4 (x - 1)} = \frac{4 (x + 2) (x - 1)}{4 x {(x - 1)}^{2}} & \begin{array}{l} Multiply \\ numerator and \\ denominator by \\ 4 (x - 1) . \end{array} \\ \frac{3 x}{4 {(x - 1)}^{2}} = \frac{(3 x) x}{4 {(x - 1)}^{2} x} = \frac{3 x^{2}}{4 x {(x - 1)}^{2}} & \begin{array}{l} Multiply numerator \\ and denominator \\ by x . \end{array} \end{array}$
3. Steps 3–4 Subtract.
  
  $\begin{array}{rcl} \frac{x + 2}{x^{2} - x} - \frac{3 x}{4 {(x - 1)}^{2}} & = & \frac{4 (x + 2) (x - 1)}{4 x {(x - 1)}^{2}} - \frac{3 x^{2}}{4 x {(x - 1)}^{2}} = \frac{4 (x + 2) (x - 1) - 3 x^{2}}{4 x {(x - 1)}^{2}} \\ = & \frac{4 x^{2} + 4 x - 8 - 3 x^{2}}{4 x {(x - 1)}^{2}} = \frac{x^{2} + 4 x - 8}{4 x {(x - 1)}^{2}} \end{array}$

Practice Problem 7

Add or subtract as indicated. Simplify your answers.
1. $\frac{4}{x^{2} - 4 x + 4} + \frac{x}{x^{2} - 4}, x \neq 2, x = - 2$
2. $\frac{2 x}{3 {(x - 5)}^{2}} - \frac{6 x}{2 (x^{2} - 5 x)}, x \neq 0, x \neq 5$

Complex Fractions

5 Identify and simplify complex fractions.

A rational expression that contains another rational expression in its numerator or denominator (or both) is called a complex rational expression or complex fraction. To simplify a complex fraction, we write it as a rational expression in lowest terms.

There are two effective methods of simplifying complex fractions.

Example 8 Simplifying a Complex Fraction

Simplify $\frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^{2} - 4}{2 x}}, x \neq 0, x \neq 2, x \neq - 2,$ using each of the two methods.

Solution

Method 1
Method 2 The LCD of $\frac{1}{2}, \frac{1}{x},$ and $\frac{x^{2} - 4}{2 x}$ is 2x.

$\begin{array}{rcl} \frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^{2} - 4}{2 x}} & = & \frac{(\frac{1}{2} + \frac{1}{x}) (2 x)}{(\frac{x^{2} - 4}{2 x}) (2 x)} \begin{array}{l} Multiply numerator and \\ denominator by the LCD . \end{array} \\ = & \frac{\frac{1}{2} (2 x) + \frac{1}{x} (2 x)}{[\frac{(x^{2} - 4)}{2 x}] (2 x)} = \frac{x + 2}{x^{2} - 4} = \frac{x + 2}{(x - 2) (x + 2)} = \frac{1}{x - 2} \end{array}$

Practice Problem 8

Simplify: $\frac{\frac{5}{3 x} + \frac{1}{3}}{\frac{x^{2} - 25}{3 x}}, x \neq 0, x \neq 5, x \neq - 5$

Example 9 Simplifying a Complex Fraction

Simplify: $\frac{x}{2 x - \frac{7}{3 - \frac{1}{2}}}, x \neq \frac{7}{5}$

Solution

We use Method 1.

Practice Problem 9

Simplify: $\frac{5 x}{3 x - \frac{4}{2 - \frac{1}{3}}}, x \neq \frac{4}{5}$

Section P.5 Exercises

Concepts and Vocabulary

The least common denominator of two rational expressions is the polynomial of least degree that contains as a factor.
The first step in finding the LCD of two rational expressions is to the denominators completely.
If the denominators of two rational expressions are $x^{2} - 2 x$ and $x^{2} - x - 2,$ the LCD is .
A rational expression that contains another rational expression in its numerator or denominator is called a(n) .
True or False. The expression $\frac{1}{2} + \frac{1}{x}$ is a complex fraction.
True or False. A polynomial is also a rational expression.
True or False. In simplifying rational expressions to lowest terms, we use $\frac{a + b}{a + c} = \frac{b}{c} .$
True or False. The fraction $\frac{2 x^{2} + 3 x}{x + 1}$ is in lowest terms.

Building Skills

In Exercises 9–24, reduce each rational expression to lowest terms. Specify the domain of the rational expression by identifying all real numbers that must be excluded from the domain.

$\frac{2 x + 2}{x^{2} + 2 x + 1}$
$\frac{3 x - 6}{x^{2} - 4 x + 4}$
$\frac{3 x + 3}{x^{2} - 1}$
$\frac{10 - 5 x}{4 - x^{2}}$
$\frac{2 x - 6}{9 - x^{2}}$
$\frac{15 + 3 x}{x^{2} - 25}$
$\frac{2 x - 1}{1 - 2 x}$
$\frac{2 - 5 x}{5 x - 2}$
$\frac{x^{2} - 6 x + 9}{4 x - 12}$
$\frac{x^{2} - 10 x + 25}{3 x - 15}$
$\frac{7 x^{2} + 7 x}{x^{2} + 2 x + 1}$
$\frac{4 x^{2} + 12 x}{x^{2} + 6 x + 9}$
$\frac{x^{2} - 11 x + 10}{x^{2} + 6 x - 7}$
$\frac{x^{2} + 2 x - 15}{x^{2} - 7 x + 12}$
$\frac{6 x^{4} + 14 x^{3} + 4 x^{2}}{6 x^{4} - 10 x^{3} - 4 x^{2}}$
$\frac{3 x^{3} + x^{2}}{3 x^{4} - 11 x^{3} - 4 x^{2}}$

In Exercises 25–42, multiply or divide as indicated. Simplify and leave the numerator and denominator in your answer in factored form. Assume all denominators are nonzero.

$\frac{x - 3}{2 x + 4} \cdot \frac{10 x + 20}{5 x - 15}$
$\frac{6 x + 4}{2 x - 8} \cdot \frac{x - 4}{9 x + 6}$
$\frac{2 x + 6}{4 x - 8} \cdot \frac{x^{2} + x - 6}{x^{2} - 9}$
$\frac{25 x^{2} - 9}{4 - 2 x} \cdot \frac{4 - x^{2}}{10 x - 6}$
$\frac{x^{2} - 7 x}{x^{2} - 6 x - 7} \cdot \frac{x^{2} - 1}{x^{2}}$
$\frac{x^{2} - 9}{x^{2} - 6 x + 9} \cdot \frac{5 x - 15}{x + 3}$
$\frac{x^{2} - x - 6}{x^{2} + 3 x + 2} \cdot \frac{x^{2} - 1}{x^{2} - 9}$
$\frac{x^{2} + 2 x - 8}{x^{2} + x - 20} \cdot \frac{x^{2} - 16}{x^{2} + 5 x + 4}$
$\frac{2 - x}{x + 1} \cdot \frac{x^{2} + 3 x + 2}{x^{2} - 4}$
$\frac{3 - x}{x + 5} \cdot \frac{x^{2} + 8 x + 15}{x^{2} - 9}$
$\frac{x + 2}{6} \div \frac{4 x + 8}{9}$
$\frac{x + 3}{20} \div \frac{4 x + 12}{9}$
$\frac{x^{2} - 9}{x} \div \frac{2 x + 6}{5 x^{2}}$
$\frac{x^{2} - 1}{3 x} \div \frac{7 x - 7}{x^{2} + x}$
$\frac{x^{2} + 2 x - 3}{x^{2} + 8 x + 16} \div \frac{x - 1}{3 x + 12}$
$\frac{x^{2} + 5 x + 6}{x^{2} + 6 x + 9} \div \frac{x^{2} + 3 x + 2}{x^{2} + 7 x + 12}$
$(\frac{x^{2} - 9}{x^{3} + 8} \div \frac{x + 3}{x^{3} + 2 x^{2} - x - 2}) \frac{1}{x^{2} - 1}$
$(\frac{x^{2} - 25}{x^{2} - 3 x - 4} \div \frac{x^{2} + 3 x - 10}{x^{2} - 1}) \frac{x - 2}{x - 5}$

In Exercises 43–60, add and subtract as indicated. Simplify and leave the numerator and denominator in your answer in factored form. Exercises 57–60 require an LCD.

$\frac{x}{5} + \frac{3}{5}$
$\frac{7}{4} - \frac{x}{4}$
$\frac{x}{2 x + 1} + \frac{4}{2 x + 1}$
$\frac{2 x}{7 x - 3} + \frac{x}{7 x - 3}$
$\frac{x^{2}}{x + 1} - \frac{x^{2} - 1}{x + 1}$
$\frac{2 x + 7}{3 x + 2} - \frac{x - 2}{3 x + 2}$
$\frac{4}{3 - x} + \frac{2 x}{x - 3}$
$\frac{- 2}{1 - x} + \frac{2 - x}{x - 1}$
$\frac{5 x}{x^{2} + 1} + \frac{2 x}{x^{2} + 1}$
$\frac{x}{2 {(x - 1)}^{2}} + \frac{3 x}{2 {(x - 1)}^{2}}$
$\frac{7 x}{2 (x - 3)} + \frac{x}{2 (x - 3)}$
$\frac{4 x}{4 {(x + 5)}^{2}} + \frac{8 x}{4 {(x + 5)}^{2}}$
$\frac{x}{x^{2} - 4} - \frac{2}{x^{2} - 4}$
$\frac{5 x}{x^{2} - 1} - \frac{5}{x^{2} - 1}$
$\frac{x - 2}{2 x + 1} - \frac{x}{2 x - 1}$
$\frac{2 x - 1}{4 x + 1} - \frac{2 x}{4 x - 1}$
$\frac{- x}{x + 2} + \frac{x - 2}{x} - \frac{x}{x - 2}$
$\frac{3 x}{x - 1} + \frac{x + 1}{x} - \frac{2 x}{x + 1}$

In Exercises 61–68, find the LCD for each pair of rational expressions.

$\frac{5}{3 x - 6}, \frac{2 x}{4 x - 8}$
$\frac{5 x + 1}{7 + 21 x}, \frac{1 - x}{3 + 9 x}$
$\frac{3 - x}{4 x^{2} - 1}, \frac{7 x}{{(2 x + 1)}^{2}}$
$\frac{14 x}{{(3 x - 1)}^{2}}, \frac{2 x + 7}{9 x^{2} - 1}$
$\frac{1 - x}{x^{2} + 3 x + 2}, \frac{3 x + 12}{x^{2} - 1}$
$\frac{5 x + 9}{x^{2} - x - 6}, \frac{x + 5}{x^{2} - 9}$
$\frac{7 - 4 x}{x^{2} - 5 x + 4}, \frac{x^{2} - x}{x^{2} + x - 2}$
$\frac{13 x}{x^{2} - 2 x - 3}, \frac{2 x^{2} - 4}{x^{2} + 3 x + 2}$

In Exercises 69–84, perform the indicated operations and simplify the result. Leave the numerator and denominator in your answer in factored form.

$\frac{5}{x - 3} + \frac{2 x}{x^{2} - 9}$
$\frac{3 x}{x - 1} + \frac{x}{x^{2} - 1}$
$\frac{2 x}{x^{2} - 4} - \frac{x}{x + 2}$
$\frac{3 x - 1}{x^{2} - 16} - \frac{2 x + 1}{x - 4}$
$\frac{x - 2}{x^{2} + 3 x - 10} + \frac{x + 3}{x^{2} + x - 6}$
$\frac{x + 3}{x^{2} - x - 2} + \frac{x - 1}{x^{2} + 2 x + 1}$
$\frac{2 x - 3}{9 x^{2} - 1} + \frac{4 x - 1}{{(3 x - 1)}^{2}}$
$\frac{3 x + 1}{{(2 x + 1)}^{2}} + \frac{x + 3}{4 x^{2} - 1}$
$\frac{x - 3}{x^{2} - 25} - \frac{x - 3}{x^{2} + 9 x + 20}$
$\frac{2 x}{x^{2} - 16} - \frac{2 x - 7}{x^{2} - 7 x + 12}$
$\frac{3}{x^{2} - 4} + \frac{1}{2 - x} - \frac{1}{2 + x}$
$\frac{2}{5 + x} + \frac{5}{x^{2} - 25} + \frac{7}{5 - x}$
$\frac{x + 3 a}{x - 5 a} - \frac{x + 5 a}{x - 3 a}$
$\frac{3 x - a}{2 x - a} - \frac{2 x + a}{3 x + a}$
$\frac{1}{x + h} - \frac{1}{x}$
$\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}$

In Exercises 85–98, perform the indicated operations and simplify the result. Leave the numerator and denominator in your answer in factored form.

$\frac{\frac{2}{x}}{\frac{3}{x^{2}}}$
$\frac{\frac{- 6}{x}}{\frac{2}{x^{3}}}$
$\frac{\frac{1}{x}}{1 - \frac{1}{x}}$
$\frac{\frac{1}{x^{2}}}{\frac{1}{x^{2}} - 1}$
$\frac{\frac{1}{x} - 1}{\frac{1}{x} + 1}$
$\frac{2 - \frac{2}{x}}{1 + \frac{2}{x}}$
$\frac{\frac{1}{x} - x}{1 - \frac{1}{x^{2}}}$
$\frac{x - \frac{1}{x}}{\frac{1}{x^{2}} - 1}$
$x - \frac{x}{x + \frac{1}{2}}$
$x + \frac{x}{x + \frac{1}{2}}$
$\frac{\frac{1}{x + h} - \frac{1}{x}}{h}$
$\frac{\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}}{h}$
$\frac{\frac{1}{x - a} + \frac{1}{x + a}}{\frac{1}{x - a} - \frac{1}{x + a}}$
$\frac{\frac{1}{x - a} + \frac{1}{x + a}}{\frac{x}{x - a} - \frac{a}{x + a}}$

Applying the Concepts

Bearing length. The length (in centimeters) of a bearing in the shape of a cylinder is given by $\frac{125.6}{π r^{2}},$ where r is the radius of the base of the cylinder. Find the length of a bearing with a diameter of 4 centimeters, using 3.14 as an approximation of $π .$
Toy box height. The height (in feet) of an open toy box that is twice as long as it is wide is given by $\frac{13.5 - 2 x^{2}}{6 x},$ where x is the width of the box. Find the height of a toy box that is 1.5 feet wide.
Diluting a mixture. A 100-gallon mixture of citrus extract and water is 3% citrus extract.
1. Write a rational expression in x whose values give the percentage (in decimal form) of the mixture that is citrus extract when x gallons of water are added to the mixture.
2. Find the percentage of citrus extract in the mixture assuming that 50 gallons of water are added to it.
Acidity in a reservoir. A half-full 400,000-gallon reservoir is found to be 0.75% acid.
1. Write a rational expression in x whose values give the percentage (in decimal form) of acid in the reservoir when x gallons of water are added to it.
2. Find the percentage of acid in the reservoir assuming that 100,000 gallons of water are added to it.
Diet lemonade. A company packages its powdered diet lemonade mix in containers in the shape of a cylinder. The top and bottom are made of a tin product that costs 5 cents per square inch. The side of the container is made of a cheaper material that costs 1 cent per square inch. The height of any cylinder can be found by dividing its volume by the area of its base.
1. Assuming that x is the radius of the base (in inches), write a rational expression in x whose values give the cost of each container, assuming that the capacity of the container is 120 cubic inches.
2. Find the cost of a container whose base is 4 inches in diameter, using 3.14 as an approximation of $π .$
Storage containers. A company makes storage containers in the shape of an open box with a square base. All five sides of the box are made of a plastic that costs 40 cents per square foot. The height of any box can be found by dividing its volume by the area of its base.
1. Assuming that x is the length of the base (in feet), write a rational expression in x whose values give the cost of each container, assuming that the capacity of the container is 2.25 cubic feet.
2. Find the cost of a container whose base is 1.5 feet long.

Beyond the Basics

In Exercises 105–114, simplify each expression.

$\frac{{(3 x + 4)}^{2} - {(2 x + 3)}^{2}}{{(3 x + 4)}^{2}}$
$\frac{(x^{2} + 1) - x (2 x)}{{(x^{2} + 1)}^{2}}$
$\frac{(x^{2} - 2 x + 1) (2 x + 2) - (x^{2} + 2 x + 1) (2 x - 2)}{{(x^{2} - 2 x + 1)}^{2}}$
$\frac{(3 x^{2} + 4 x + 1) (4 x + 1) - (2 x^{2} + x - 1) (6 x + 4)}{{(3 x^{2} + 4 x + 1)}^{2}}$
$\frac{x^{2} - x - 20}{x^{2} - 25} \cdot \frac{x^{2} - x - 2}{x^{2} + 2 x - 8} \div \frac{x + 1}{x^{2} + 5 x}$
$\frac{y^{4} - x^{4}}{y^{2} - 2 x y + x^{2}} \div (\frac{y^{2} x + x^{3}}{y^{3} - x^{3}} \cdot \frac{y^{4} + y^{2} x^{2} + x^{4}}{y^{2} x - y x^{2} + x^{3}})$
$\frac{1}{a - \frac{1}{a + \frac{1}{a}}} - \frac{1}{a + \frac{1}{a - \frac{1}{a}}}$
$\frac{1 + \frac{x - y}{x + y}}{1 - \frac{x - y}{x + y}} \div \frac{1 + \frac{x^{2} - y^{2}}{x^{2} + y^{2}}}{1 - \frac{x^{2} - y^{2}}{x^{2} + y^{2}}}$
$(\frac{x^{2}}{1 - x^{4}} + \frac{2 x^{4}}{1 - x^{8}}) \div \frac{x^{2} + 1}{x}$
$\frac{\frac{x^{2} + y^{2}}{y} - x}{\frac{1}{y} - \frac{1}{x}} \cdot \frac{x^{2} - y^{2}}{x^{3} + y^{3}}$

Getting Ready for the Next Section

In Exercises 115–120, state whether the statement is True or False.

$2 \sqrt{3} + 5 \sqrt{3} = (2 + 5) \sqrt{3} = 7 \sqrt{3}$
$x \sqrt{2} + y \sqrt{2} - z \sqrt{2} = (x + y - z) \sqrt{2}$
$\sqrt{{(- 4)}^{2}} = - 4$
${(\sqrt{4})}^{2} = 4$
$\sqrt{4 \cdot 16} = \sqrt{4} \cdot \sqrt{16}$
${(a + b)}^{2} = a^{2} + b^{2}$

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Table of Contents for Section P.5 Rational Expressions

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Table of Contents for
Section P.5 Rational Expressions