# Rational Expressions and Equations

## Use Direct and Inverse Variation

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.

### Solve Direct Variation Problems

Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation
Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.
In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.
We’ll list the steps below.
Now we’ll solve a few applications of direct variation.
In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.
In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is . We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k.

### Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is .
The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is .
We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

### Section Exercises


### Practice Makes Perfect

Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
Write an inverse variation equation to solve the following problems.

### Mixed Practice


### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

### Chapter 8 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.
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 and Divide Rational Expressions

Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.

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

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.

### Add and Subtract Rational Expressions With Unlike Denominators

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 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.

### Solve Rational Equations

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 Proportion and Similar Figure Applications Similarity

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 and Work Applications Problems

Solve Uniform Motion Applications
In the following exercises, solve.
Solve Work Applications
In the following exercises, solve.

### Use Direct and Inverse Variation

Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.

### 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.