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Elementary Algebra
Rational Expressions

1
Reducing Rational Expressions to Lowest Terms

Problem  1

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

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Breylor cc
Breylor
Nathan cc
Nathan
Problem  3
practice icon

State the restrictions on the variable \(\displaystyle\frac{x+2}{x-3}\)

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Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4

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

Choose instructor to watch:
CJ cc
CJ
Stefanie cc
Stefanie
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  5
practice icon

Reduce \(\displaystyle\frac{x^2-9}{x^2+5x+6}\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
Edwin espanol spanish
Edwin
Problem  6

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

Choose instructor to watch:
Mrs. Mooney cc
Mrs. Mooney
Breylor cc
Breylor
Nathan cc
Nathan
Problem  7
practice icon

Reduce \(\displaystyle\frac{2x^3+2x^2-24x}{x^3+2x^2-8x}\)

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Jesse cc
Jesse
Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  8

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

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Breylor cc
Breylor
Nathan cc
Nathan
Problem  9
A solution of hydrochloric acid (HCl) and water contains 49 milliliters of water and 21 milliliters of HCl. Find the ratio of HCl to water and of HCl to the total volume of the solution.
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Lauren cc
Lauren
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  10

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?

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Breylor cc
Breylor
Nathan cc
Nathan
Mini Lecture

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. \(\displaystyle\frac{4x^2-12x+9}{4x^2-9}\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

2
Multiplication and Division of Rational Expressions

Problem  1
practice icon

Multiply \(\displaystyle\frac{x-2}{x+3}\cdot \displaystyle\frac{x^2-9}{2x-4}\)

Choose instructor to watch:
Betsy cc
Betsy
Molly M. cc
Molly M.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
practice icon

Multiply \(\displaystyle\frac{3a+6}{a^2}\cdot \displaystyle\frac{a}{2a+4}\)

Choose instructor to watch:
Betsy cc
Betsy
Molly M. cc
Molly M.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  3
practice icon

Divide \(\displaystyle\frac{x^2+7x+12}{x^2-16}\div \displaystyle\frac{x^2+6x+9}{2x-8}\)

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Betsy cc
Betsy
Molly M. cc
Molly M.
Preston cc
Preston
David espanol spanish
David
Problem  4

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

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Julieta cc
Julieta
Breylor cc
Breylor
Nathan cc
Nathan
Problem  5

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

Choose instructor to watch:
Breylor cc
Breylor
Nathan cc
Nathan
Problem  6
practice icon

Multiply \(\; a(a+5)(a-5)\left(\displaystyle\frac{a+4}{a^2+5a}\right)\)

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Betsy cc
Betsy
Molly M. cc
Molly M.
CJ cc
CJ
David espanol spanish
David
Problem  7
practice icon

The Mall of America covers \(78\) acres of land. If one square mile \(=640\) acres, how many square miles does the Mall of America cover? Round to the nearest hundredth of a square mile.

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Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  8

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?

Choose instructor to watch:
Spencer cc
Spencer
Breylor cc
Breylor
Mini Lecture

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

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Mr. McKeague cc
Mr. McKeague

3
Addition and Subtraction of Rational Expressions

Problem  1
practice icon

Add \(\displaystyle\frac{5}{x}+\displaystyle\frac{3}{x}\)

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Preston cc
Preston
Molly S. cc
Molly S.
Edwin espanol spanish
Edwin
Problem  2
practice icon

Add \(\displaystyle\frac{x}{x^2-9}+\displaystyle\frac{3}{x^2-9}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Cynthia espanol spanish
Cynthia
Problem  3

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

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Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  4

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

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Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  5

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

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Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  6
practice icon

Add \(\displaystyle\frac{1}{10}+\displaystyle\frac{3}{14}\)

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Preston cc
Preston
Molly S. cc
Molly S.
Edwin espanol spanish
Edwin
Problem  7
practice icon

Subtract \(\displaystyle\frac{3}{x}-\displaystyle\frac{1}{2}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Cynthia espanol spanish
Cynthia
Problem  8
practice icon

Add \(\displaystyle\frac{5}{2x-6}+\displaystyle\frac{x}{x-3}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Edwin espanol spanish
Edwin
Problem  9
practice icon

Add \(\displaystyle\frac{1}{x+4}+\displaystyle\frac{8}{x^2-16}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Cynthia espanol spanish
Cynthia
Problem  10

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

Choose instructor to watch:
Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  11
practice icon

Subtract \(\displaystyle\frac{x+4}{2x+10}-\displaystyle\frac{5}{x^2-25}\)

Choose instructor to watch:
Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  12
Write an expression for the sum of a number and its reciprocal, and then simplify that expression.
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Betsy cc
Betsy
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Mini Lecture

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. \(\displaystyle\frac{8}{x^2-16}-\displaystyle\frac{7}{x^2-x-12}\)

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Mr. McKeague cc
Mr. McKeague

4
Complex Fractions

Problem  1
practice icon

Simplify \(\displaystyle\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{2}{3}}\)

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Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
practice icon

Simplify \(\displaystyle\frac{\displaystyle\frac{2x^3}{y^2}}{\displaystyle\frac{4x}{y^5}}\)

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  3
practice icon

Simplify \(\displaystyle\frac{x+\displaystyle\frac{1}{y}}{y+\displaystyle\frac{1}{x}}\)

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
practice icon

Simplify \(\displaystyle\frac{1-\displaystyle\frac{4}{x^2}}{1-\displaystyle\frac{1}{x}-\displaystyle\frac{6}{x^2}}\)

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Molly S. cc
Molly S.
Preston cc
Preston
Cynthia espanol spanish
Cynthia
Problem  5

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

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Mini Lecture

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. \(\displaystyle\frac{1-\frac{9}{y^2}}{1-\frac{1}{y}-\frac{6}{y^2}}\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague
Take 5

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.

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Mr. McKeague cc
Mr. McKeague

5
Equations Involving Rational Expressions

Problem  1
practice icon

Solve \(\displaystyle\frac{x}{3}+\displaystyle\frac{5}{2}=\displaystyle\frac{1}{2}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
David espanol spanish
David
Problem  2
practice icon

Solve \(\displaystyle\frac{3}{x-1}=\displaystyle\frac{3}{5}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Edwin espanol spanish
Edwin
Problem  3

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

Choose instructor to watch:
Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  4
practice icon

Solve \(\displaystyle\frac{x}{x^2-9}-\displaystyle\frac{3}{x-3}=\displaystyle\frac{1}{x+3}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Edwin espanol spanish
Edwin
Problem  5
practice icon

Solve \(\displaystyle\frac{x}{x-3}+\displaystyle\frac{3}{2}=\displaystyle\frac{3}{x-3}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
Cynthia espanol spanish
Cynthia
Problem  6
practice icon

Solve \(\displaystyle\frac{a+4}{a^2+5a}=\displaystyle\frac{-2}{a^2-25}\)

Choose instructor to watch:
Preston cc
Preston
Molly S. cc
Molly S.
Betsy cc
Betsy
David espanol spanish
David
Mini Lecture

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

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

6
Proportions

Problem  1
practice icon

Solve \(\displaystyle\frac{3}{x}=\displaystyle\frac{6}{7}\)

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Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
practice icon

Solve \(\displaystyle\frac{x+1}{2}=\displaystyle\frac{3}{x}\)

Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Cynthia
Cynthia
Problem  3

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?

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Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
A woman drives her car 270 miles in 6 hours. If she continues at the same rate, how far will she travel in 10 hours?
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Mr. McKeague cc
Mr. McKeague
Preston cc
Preston
Arielle cc
Arielle
David cc espanol spanish
David
Problem  5
A baseball player gets 8 hits in the first 18 games of the season. If he continues at the same rate, how many hits will he get in 45 games?
Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Mini Lecture

Mini Lecture
Solve.

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

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

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

7
Applications

Problem  1
One number is twice another. The sum of their reciprocals is 9/2. Find the two numbers.
Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
A boat travels 30 miles up a river in the same amount of time it takes to travel 50 miles down the river. If the current is 5 miles per hour, what is the speed of the boat in still water?
Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Problem  3
Tina is training for a biathlon. She rides her bike 15 miles up a hill and then 15 miles back down the same hill. The complete trip takes her 2 hours. If her downhill speed is 20 miles per hour faster than her uphill speed, how fast does she ride uphill?
Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4

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?

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Julieta cc
Julieta
Spencer cc
Spencer
Breylor cc
Breylor
Problem  5
An inlet pipe can fill a water tank in 10 hours, while an outlet pipe can empty the same tank in 15 hours. By mistake, both pipes are left open. How long will it take to fill the water tank with both pipes open?
Choose instructor to watch:
Stefanie cc
Stefanie
Preston cc
Preston
Edwin espanol spanish
Edwin
Mini Lecture

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?

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

8
Variation

Problem  1

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

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Stephanie cc
Stephanie
Winston cc
Winston
Breylor cc
Breylor
Problem  2

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

Choose instructor to watch:
Stephanie cc
Stephanie
Winston cc
Winston
Breylor cc
Breylor
Problem  3

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

Choose instructor to watch:
Stephanie cc
Stephanie
Winston cc
Winston
Breylor cc
Breylor
Problem  4

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

Choose instructor to watch:
Stephanie cc
Stephanie
Winston cc
Winston
Breylor cc
Breylor
Problem  5
practice icon

Suppose \(y\) varies directly as \(x\). If \(y\) is \(15\), when \(x\) is \(3\), find \(y\) when \(x\) is \(4\).

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6

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

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  7

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

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Stephanie cc
Stephanie
Breylor cc
Breylor
Winston cc
Winston
Problem  8

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

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Stephanie cc
Stephanie
Breylor cc
Breylor
Problem  9

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

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Stephanie cc
Stephanie
Breylor cc
Breylor
Winston cc
Winston
Problem  10

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

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Winston cc
Winston
Breylor cc
Breylor
Stephanie cc
Stephanie
Problem  11
practice icon

Suppose \(y\) varies inversely as \(x\). When \(y\) is \(4\), \(x\) is \(5\). Find \(y\) when \(x\) is \(10\).

Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  12
The cost of a certain kind of candy varies directly with the weight of the candy. If 12 ounces of the candy cost $1.68, how much will 16 ounces cost?
Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  13
The intensity of light from a source varies inversely as the square of the distance from the source. Ten feet away from the source the intensity is 200 candlepower. What is the intensity 5 feet from the source?
Choose instructor to watch:
Molly S. cc
Molly S.
Preston cc
Preston
Edwin espanol spanish
Edwin
Mini Lecture

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 \(I=7\).

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague