What are the similarities and differences between these two expressions and how we evaluate them? \text{Expression 1: } \ \frac{1}{7}-\frac{1}{14} \qquad \text{Expression 2: }\ \frac{1}{7}+\frac{3}{7}

Create an expression in the form: \frac{â¬š}{â¬š} + \frac{â¬š}{â¬š} where each blank is filled with a unique, nonzero integer value.

Rewrite your original expression into two new expressions by:

Multiplying one term by \dfrac{1}{x}, where x is a positive integer, resulting in: \frac{â¬š}{â¬šx} + \frac{â¬š}{â¬š} \text{\quad or \quad} \frac{â¬š}{â¬š} + \frac{â¬š}{â¬šx}

Adding x to the denominator of each fraction, resulting in: \frac{â¬š}{x+ â¬š} + \frac{â¬š}{x + â¬š}

Work with a partner to determine how to add the fractions created in step 3.

The sum of two **rational expressions** will result in another rational expression. Recall that a common denominator is required in order to add or subtract fractions. The same is true for rational expressions A, B, and C:\frac{A}{B} + \frac{C}{B} = \frac{A+C}{B}

In order to add or subtract rational expressions which have different denominators, we will need to find a **common multiple** to rewrite the expressions so that they share a common denominator. Given \dfrac{A}{B} + \dfrac{C}{D} where A, B, C, and D are expressions, common multiple is B\cdot D, so we have:

\displaystyle \frac{A}{B} + \frac{C}{D} | \displaystyle = | \displaystyle \frac{A}{B} \cdot \frac{D}{D} + \frac{C}{D} \cdot \frac{B}{B} | Multiplicative identity, since \dfrac{D}{D}= \dfrac{B}{B}=1 |

\displaystyle = | \displaystyle \frac{AD}{BD} + \frac{CB}{DB} | Multiply the fractions | |

\displaystyle = | \displaystyle \frac{AD}{BD} + \frac{BC}{BD} | Commutative property of multiplication | |

\displaystyle = | \displaystyle \frac{AD + BC}{BD} | Add the fractions with a common denominator |

We need to state restrictions on the variables so we do not get an expression with 0 in the denominator, leading to an undefined expression.

Fully simplify the expression, justifying each step. Write any restrictions on the variables.\frac{k - 4}{3 k} - \frac{k - 22}{3 k}

Worked Solution

Determine whether the two expressions are equivalent, justifying your answer.

Expression 1:\frac{3x}{\left(x-2\right)} + \frac{\left(-2x+1\right)}{5x}

Expression 2:\frac{3x}{\left(x-2\right)} - \frac{\left(2x+1\right)}{5x}

Worked Solution

Fully simplify the rational expressions, justifying each step. State any restrictions on the variables.

a

\frac{5m}{2p^{5}} + \frac{4}{p^{2}m^{2}}

Worked Solution

b

\frac{y - 2}{6} + \frac{y + 3}{y + 9}

Worked Solution

Fully simplify the expression, justifying each step. State any restrictions on the variables.\frac{2x + 5}{x^{2} - 2x - 3} - \frac{x}{x^{2} - 6x + 9}

Worked Solution

Fully simplify the expression, justifying each step. State any restrictions on the variables.

\frac{ \frac{4}{x+3} + 6}{ 2 + \frac{2}{x+3}}

Worked Solution

Idea summary

Prior to adding or subtracting rational expressions, do the following:

- Determine restrictions on the variables that will lead to undefined expressions
- If necessary, rewrite rational expressions to get a common denominator, using the multiplicative identity property: \dfrac{A}{A}=1 for any rational expression, A where A\neq 0.