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

In the introduction to this chapter, we saw that the average cost per check of writing \(x\) checks in one month is given by the rational expression \[A=\dfrac{2.00+0.15x}{x}\] Find the average cost per check if \(8\) checks are written in a month.

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 \(I=7\).