### Key Concepts

#### 7.1 Multiply and Divide Rational Expressions

**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 $\frac{a}{b}=\frac{a\xb7c}{b\xb7c}$ and $\frac{a\xb7c}{b\xb7c}=\frac{a}{b}.$**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 $b-a.$

$\frac{a-b}{b-a}=\mathrm{-1}\phantom{\rule{8em}{0ex}}a\ne b$

An expression and its opposite divide to $\mathrm{-1}.$**Multiplication of Rational Expressions**

If*p*,*q*,*r*, and*s*are polynomials where $q\ne 0,s\ne 0,$ then

$\phantom{\rule{8em}{0ex}}\frac{p}{q}\xb7\frac{r}{s}=\frac{pr}{qs}$**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

$\phantom{\rule{8em}{0ex}}\frac{p}{q}\xf7\frac{r}{s}=\frac{p}{q}\xb7\frac{s}{r}$**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.

#### 7.2 Add and Subtract Rational Expressions

**Rational Expression Addition and Subtraction**

If*p*,*q*, and*r*are polynomials where $r\ne 0,$ then

$\phantom{\rule{8em}{0ex}}\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$ 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.

- Step 1.
Determine if the expressions have a common denominator.

#### 7.3 Simplify Complex Rational Expressions

**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.

#### 7.4 Solve Rational Equations

**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.

#### 7.5 Solve Applications with Rational Equations

- 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 $\text{\Delta}ABC$ is similar to $\text{\Delta}XYZ,$ 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 $y.$

- For any two variables
**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*.

- For any two variables

#### 7.6 Solve Rational Inequalities

**Solve a rational inequality.**- Step 1. Write the inequality as one quotient on the left and zero on the right.
- Step 2. Determine the critical points–the points where the rational expression will be zero or undefined.
- Step 3. Use the critical points 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.