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Adding and Subtracting Rational Expressions Solutions

Rational expressions are closed under addition, subtraction, multiplication, and division. 
Mathematically, closed means you start and end a problem with the same type of expression. 
Therefore, if you start with a rational expression, you end with rational expressions .

Note

Recall that a rational expression is the ratio of p to q, where p and q are polynomial expressions.

For example: 
- The sum of rational expressions is always a rational expression. 
- The difference between rational expressions is always a rational expression. 
- The product or quotient of rational expressions is always a rational expression. 
When adding or subtracting rational expressions , determine a least common denominator (LCD) and use it prior to finding the sum or difference.
Follow these guidelines when simplifying rational expressions by adding or subtracting: 
1. 1. Factor the denominators to determine the LCD.  
  2. Rewrite each term in the rational expression using the LCD (Identity Property of Multiplication).  
  3. Simplify and combine the numerators to form one expression .

Naming the excluded values (restrictions) is best practice because it gives the most complete answer to a simplified expression.

Example 1

Simplify.

$\frac{x^{2} + 3 x}{x^{2} + 7 x + 6} - \frac{4 x - 11}{x^{2} + 7 x + 6}$

Plan 
Factor denominators to determine the LCD 

Name the restrictions for the denominator 

Simplify the numerator 

Write the answer as a fraction, including restrictions

Implement

$\frac{x^{2} + 3 x}{(x + 1) (x + 6)} - \frac{4 x - 11}{(x + 1) (x + 6)}$

$x \neq - 6, - 1$

$x^{2} + 3 x - (4 x - 11) x^{2} + 3 x - 4 x + 11 x^{2} - x + 11$

$\frac{x^{2} - x + 11}{(x + 1) (x + 6)}, x \neq - 6, - 1$

Explain

Factor the denominators
Name the restrictions

Note

You should be able to find the restrictions using mental math.

Simplify the numerator

Note

Once you find the LCD, you only need to simplify the numerator.

It is not necessary to write the denominator while simplifying the numerator.

Write the answer with the factored denominator and restrictions

Note

See Lesson 3 More to Explore for ways to use technology to check answers and determine restrictions.

Example 2

Simplify.

$\frac{2 x^{2} - 5}{x^{2} - 13 x + 12} + \frac{x - 7}{x - 12} - \frac{3}{x^{2} - x}$

Plan
Factor all denominators to determine the LCD

Write each expression with LCD

Simplify

Implement

$\frac{2 x^{2} - 5}{(x - 1) (x - 12)} + \frac{x - 7}{x - 12} - \frac{3}{x (x - 1)}$

$LCD : x (x - 1) (x - 12) x \neq 0, 1, 12$

$\frac{x (2 x^{2} - 5)}{x (x - 1) (x - 12)} + \frac{x (x - 1) (x - 7)}{x (x - 1) (x - 12)} - \frac{3 (x - 12)}{x (x - 1) (x - 12)}$

$2 x^{3} - 5 x + x (x^{2} - 8 x + 7) - 3 x + 36 2 x^{3} - 5 x + x^{3} - 8 x^{2} + 7 x - 3 x + 36 3 x^{3} - 8 x^{2} - x + 36 \frac{3 x^{3} - 8 x^{2} - x + 36}{x (x - 1) (x - 12)}, x \neq 0, 1, 12$

Explain

Factor the denominators
Name the LCD and excluded values
Identity property
Write each expression with the LCD

Note

Remember, do not simplify (cross out) identical binomials when writing fractions with an LCD. If you simplify out terms, the result will be the given problem.

Simplify the numerator
Write the answer with a factored denominator and excluded values

Example 3

Simplify.

$\frac{2 n - 3}{n - 3} + \frac{n + 2}{(3 - n) (2 n + 3)}$

$\frac{2 n - 3}{n - 3} + \frac{n + 2}{- (n - 3) (2 n + 3)} LCD : (2 n + 3) (n - 3) x \neq - \frac{3}{2}, 3 \frac{(2 n + 3) (2 n - 3)}{(2 n + 3) (n - 3)} + \frac{- (n + 2)}{(n - 3) (2 n + 3)} 4 n^{2} - 9 - n - 2 \frac{4 n^{2} - n - 11}{(2 n + 3) (n - 3)}, x \neq - \frac{3}{2}, 3$

Note

Write all expressions in standard form. Factor $- 1$ from the expression (3 – n).

Move the negative to the numerator. The negative can be in either the numerator or denominator of an expression.

If $- 1$ was written as part of the LCD rather than moved to the numerator of the second fractional expression, the work will look slightly different.

$L C D : (- 1) (2 n + 3) (n - 3) \frac{(- 1) (2 n + 3) (2 n - 3)}{(- 1) (2 n + 3) (n - 3)} + \frac{(n + 2)}{(- 1) (n - 3) (2 n + 3)} - 4 n^{2} + 9 + n + 2 \frac{- 4 n^{2} + n + 11}{(- 1) (2 n + 3) (n - 3)}, x \neq - \frac{3}{2}, 3$

This is the same answer when the negative is distributed across the numerator.