Lesson 5: Chapter Summary and Review

New Concept

This is the chapter 'boss level'! We'll recap our journey with algebraic fractions: simplifying, operating, and solving equations.

What’s next

Ready to prove your skills? A full set of review questions and problems awaits to challenge and solidify your chapter mastery.

Property

1. Factor numerator and denominator completely.
2. Divide numerator and denominator by any common factors. We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

Problem: Reduce the fraction $\frac{x^2 + 3x + 2}{x^2 - 4}$.

Step 1: Factor everything.
The numerator factors into $(x+1)(x+2)$.
The denominator is a difference of squares, factoring into $(x-2)(x+2)$.

Property

The opposite of $a - b$ is

Examples

Problem: Simplify the fraction $\frac{y - 4}{16 - y^2}$.

Step 1: Factor the denominator.
$16 - y^2 = (4-y)(4+y)$. The fraction is $\frac{y - 4}{(4-y)(4+y)}$.

Property

1. Factor each numerator and denominator completely.
2. If any factor appears in both a numerator and a denominator, divide out that factor.
3. Multiply the remaining factors.

Examples

Problem: Multiply $\frac{x^2-9}{5x} \cdot \frac{10}{x+3}$.

Step 1: Factor all parts.
The first numerator is $x^2-9 = (x-3)(x+3)$. The fraction is $\frac{(x-3)(x+3)}{5x} \cdot \frac{10}{x+3}$.

Property

1. Take the reciprocal of the second fraction and change the division to multiplication.
2. Follow the rules for multiplication of fractions.

Examples

Problem: Divide $\frac{z^2-4}{z+1} \div \frac{z+2}{z^2-1}$.

Step 1: Keep, Change, Flip.
Rewrite the problem as multiplication: $\frac{z^2-4}{z+1} \cdot \frac{z^2-1}{z+2}$.

Property

1. Factor each denominator completely.
2. For each factor, circle the most copies of that factor in any single denominator.
3. The LCD is the product of all the circled factors.

Examples

Problem: Find the LCD for $\frac{3}{x^2+6x+9}$ and $\frac{5}{x^2-9}$.

Step 1: Factor each denominator.
Denominator 1: $x^2+6x+9 = (x+3)(x+3) = (x+3)^2$.
Denominator 2: $x^2-9 = (x-3)(x+3)$.

Property

1. Find the lowest common denominator (LCD).
2. Build each fraction to an equivalent one with the LCD.
3. Add or subtract the numerators, and keep the same denominator.
4. Reduce if necessary.

Examples

Problem: Calculate $\frac{4}{x-3} + \frac{2}{x+1}$.

Step 1: Find the LCD.
The denominators are prime, so the LCD is $(x-3)(x+1)$.

Property

Whenever we multiply an equation by an expression containing the variable, we should check for extraneous solutions. An extraneous solution is a result that is not a valid solution to the original equation because it makes a denominator equal to zero.

Examples

Problem: Solve the equation $\frac{x}{x-5} - 2 = \frac{5}{x-5}$.

Step 1: Clear the denominator.
The LCD is $(x-5)$. Multiply every term by it: $x - 2(x-5) = 5$.