Evaluate $\dfrac{x^2-6x+9}{x^2-4}$ for $x=3$ and $x=-2$.

Determine any values of the variable for which the rational expression $\dfrac{5}{x^2-x-6}$ will be defined.

Reduce to lowest terms: $\dfrac{10a+20}{20-5a^2}$

Reduce $\dfrac{x-5}{x^2-25}$ to lowest terms. Also, state any restrictions on the variable.

The Comstock Express chair lift at the Northstar California ski resort in Lake Tahoe is $5\text{,}649$ feet long. If a ride on this chair lift takes $6$ minutes, what is the average speed of the lift in feet per minute?

Mini Lecture
Reduce to lowest terms.

1. $\displaystyle\frac{10a+20}{5a+10}$

2. $\displaystyle\frac{x^3+3x^2-4x}{x^3-16x}$

3. $\displaystyle\frac{4x^2-12x+9}{4x^2-9}$

Divide: $$\left(\dfrac{3x-9}{x^2-x-20}\right)\div\left(\dfrac{15-2x-x^2}{x^2-25}\right)$$

Multiply: $(49-x^2)\left(\dfrac{x+4}{x-7}\right)$

The Comstock Express chair lift at the Northstar California ski resort in Lake Tahoe is $5\text{,}649$ feet long. If a ride on this chair lift takes $6$ minutes, what is the average speed of the lift in feet per hour?

Mini Lecture
Multiply or divide as indicated.

1. $\frac{2x+10}{x^2}\cdot \frac{x^3}{4x+20}$

2. $\frac{y^2-5y+6}{2y+4}\div \frac{2y-6}{y+2}$

3. $\frac{4y^2-12y+9}{y^2-36}\div \frac{2y^2-5y+3}{y^2+5y-6}$

4. $\left(x^2-9\right)\left(\frac{2}{x+3}\right)$

Subtract: $\dfrac{6x-13}{x^2-2x-3}-\dfrac{2x-1}{x^2-2x-3}$

Find the least common denominator for the rational expressions $\dfrac{2}{x}$ and $\dfrac{4}{x-3}$.

Find the least common denominator for the rational expressions $$\frac{x-5}{2x^2+4x+2}\text{ and }\frac{3x}{2x^2-2}$$

Subtract: $\dfrac{2x-1}{x^2+5x+6}-\dfrac{x+1}{x^2-9}$

Mini Lecture
Combine.

1. $\displaystyle\frac{y^2}{y-1}-\displaystyle\frac{1}{y-1}$

2. $\displaystyle\frac{x}{x+1}+\displaystyle\frac{3}{4}$

3. $\displaystyle\frac{8}{x^2-16}-\displaystyle\frac{7}{x^2-x-12}$

Simplify each term and explain how this sequence is related to the Fibonacci sequence: $1+\displaystyle\frac{1}{1+1}, 1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{1+1}},1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{1+1}}}, \ldots$

Mini Lecture
Simplify.

1. $\displaystyle\frac{\frac{3}{4}}{\frac{1}{8}}$

2. $\displaystyle\frac{y+\frac{1}{x}}{x+\frac{1}{y}}$

3. $\displaystyle\frac{\frac{1}{a+2}}{\frac{1}{a^2-a-6}}$

4. $\displaystyle\frac{1-\frac{9}{y^2}}{1-\frac{1}{y}-\frac{6}{y^2}}$

Take-Five: A Continued Fraction

If you take complex fractions one step further, you get continued fractions. Here is a simple continued fraction, and a connection to the Fibonacci sequence.

Solve: $1+\dfrac{6}{x^2}=\dfrac{5}{x}$

Mini Lecture
Solve.

1. $\displaystyle\frac{x}{3}+\displaystyle\frac{1}{2}=-\displaystyle\frac{1}{2}$

2. $\displaystyle\frac{1}{y}-\displaystyle\frac{1}{2}=-\displaystyle\frac{1}{4}$

3. $1-\displaystyle\frac{8}{x}=-\displaystyle\frac{15}{x^2}$

4. $\displaystyle\frac{8}{x^2-4}+\displaystyle\frac{3}{x+2}=\displaystyle\frac{1}{x-2}$

A manufacturer knows that 8 out of every 100 parts will be defective. If the machine produces 1,450 parts, how many can be expected to be defective?

Mini Lecture
Solve.

1. $\displaystyle\frac{x}{2}=\displaystyle\frac{6}{12}$

2. $\displaystyle\frac{x+1}{3}=\displaystyle\frac{4}{x}$

Allison can clean the house in $5$ hours. Working together, she and Kaitlin can clean the house in $3$ hours. How long would it take Kaitlin, working alone, to clean the house?

Mini Lecture

1. One number is three times as large as another. The sum of their reciprocals is $16/3$. Find the numbers.

2. One plane can travel $20$ mph faster than another. One of them goes $285$ miles in the same time it takes the other to go $255$ miles. What are their speeds?

3. An inlet pipe can fill a pool in $12$ hours, while an outlet pipe can empty it in $15$hours. If both pipes are left open, how long will it take to fill the pool?

For the following direct variation statement, give an equivalent algebraic equation: $$y\text{ varies directly as }x$$

For the following direct variation statement, give an equivalent algebraic equation: $$y\text{ varies directly as the square of }x$$

For the following direct variation statement, give an equivalent algebraic equation: $$s\text{ varies directly as the square root of }t$$

For the following direct variation statement, give an equivalent algebraic equation: $$r\text{ varies directly as the cube of }s$$

Suppose $y$ varies directly as the square of $x$. When $x$ is $4$, $y$ is $32$. Find $x$ when $y$ is $50$.

For the following inverse variation statement, give an equivalent algebraic equation: $$y\text{ varies inversely as }x$$

For the following inverse variation statement, give an equivalent algebraic equation: $$y\text{ varies inversely as the square of }x$$

For the following inverse variation statement, give an equivalent algebraic equation: $$F\text{ varies inversely as the square root of }t$$

For the following inverse variation statement, give an equivalent algebraic equation: $$r\text{ varies inversely as the cube of }s$$

Mini Lecture

1. $y$ varies directly as $x$. If $y=39$ when $x=3$, find $y$ when $x$ is $10$.

2. $y$ varies inversely as the square of $x$. If $y=4$ when $x=3$, find $y$ when $x$ is $2$.

3. The power $P$ is an electric circuit varies directly with the square of the current $I$. If $P=30$ when $I=2$ find $P$ when $I=7$.