# Rational Expressions and Functions

## Solve Rational Inequalities


### Solve Rational Inequalities

We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.
Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.
Inequalities such as  and  are rational inequalities as they each contain a rational expression.
When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.
Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.
When we solve an equation and the result is  we know there is one solution, which is 3.
When we solve an inequality and the result is  we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.
To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.
Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.
We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.
We write the solution in interval notation being careful to determine whether the endpoints are included.
We summarize the steps for easy reference.
The next example requires that we first get the rational inequality into the correct form.
In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.
The next example requires some work to get it into the needed form.

### Solve an Inequality with Rational Functions

When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.
In economics, the function  is used to represent the cost of producing x units of a commodity. The average cost per unit can be found by dividing  by the number of items  Then, the average cost per unit is 

### Key Concepts

1. Solve a rational inequality.




### Section Exercises


### Practice Makes Perfect

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.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter 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.
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

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  where  and  are given.
In the following exercises, find  where  and  are given.

### Simplify Complex Rational Expressions

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.

### 7.4 Solve Rational Equations

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 Applications with Rational Equations

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

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

### Practice Test

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  given  and 
In the following exercises, solve.