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College Algebra
Algebra Review

1
The Real Number System

Problem  1
practice icon

What property does each of the following equations illustrate?

  1. \(4+6=6+4\)

  2. \(3(5-8)=3(5)-3(8)\)

  3. \(2\cdot (3\cdot 5)=(2\cdot 3)\cdot 5\)

  4. \(4\cdot 1=4\)

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2

Let \(A=\{-2,-1,1,4,5\}\), \(B=\{-3,-2,4,7,9\}\) and \(C=\{-3,0,2,7\}\). Find the following:

  1. \(B\cup C\)

  2. \(A\cap B\)

  3. \(A\cap C\)

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Julieta cc
Julieta
Mr. Galletti cc
Mr. Galletti
Julieta cc espanol spanish
Julieta
Problem  3

Graph each of the following intervals on the number line and give a verbal description.

  1. \((-2, 4]\)

  2. \((-\infty, 4]\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4

Give the corresponding interval notation for the graph of an inequality.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5

Find the indicated set, and graph.

  1. \((1,4)\cup[2,5]\)

  2. \((1,4)\cap[2,5]\)

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Julieta cc
Julieta
Mr. Galletti cc
Mr. Galletti
Julieta cc espanol spanish
Julieta
Problem  6
practice icon

Evaluate the following.

  1. \(\lvert -3\rvert\)

  2. \(-4-\lvert 4.5\rvert\)

  3. \(\lvert -4+9\rvert +3\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  7

Find the distance between \(-6\) and \(4\) on the real number line.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  8
practice icon

Simplify each expression.

  1. \(3^2+2(-3+7)\)

  2. \(\dfrac{3^2+1}{4-7}\)

  3. \(6+3(5-(-3^2+2))\)

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Mr. Hampton cc
Mr. Hampton
Gordon cc
Gordon
Saba cc
Saba
Problem  9
practice icon

Evaluate the following expression on your calculator. Check the calculator output with a hand calculation.

  1. \(4+10\div 5+2\)

  2. \(\dfrac{4+10}{5+2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  10
practice icon

Write each of the following in symbols:

The sum of \(x\) and \(5\) is less than \(2\).
The product of \(3\) and \(x\) is \(21\).
The quotient of \(y\) and \(6\) is \(4\).
Twice the difference of \(b\) and \(7\) is greater than \(5\).
The difference of twice \(b\) and \(7\) is greater than \(5\).

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Gordon cc
Gordon
Cynthia espanol spanish
Cynthia
Problem  11
practice icon

Simplify: \(\; 5+3(2+4)\)

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Aaron cc
Aaron
Lauren
Lauren
Anthony espanol spanish
Anthony
Problem  12
practice icon

Simplify: \(40-20 \div 5+8\)

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Katrina cc
Katrina
Brooke cc
Brooke
Logan cc
Logan
Anthony espanol spanish
Anthony
Problem  13
practice icon

Simplify \((-2-3)(5-9)\)

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Preston cc
Preston
Stefanie cc
Stefanie
Anthony espanol spanish
Anthony
Problem  14
practice icon

Simplify \(2-5(7-4)-6\)

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Preston cc
Preston
Stefanie cc
Stefanie
Winston cc
Winston
Julieta espanol spanish
Julieta
Problem  15
practice icon

Simplify \(2(4-7)^3+(-2-3)^2\)

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Preston cc
Preston
Stefanie cc
Stefanie
Anthony espanol spanish
Anthony
Problem  16
practice icon

Simplify: \(\dfrac{-8-8}{-5-3}\)

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Mr. McKeague cc
Mr. McKeague
Ana
Ana
Matt cc
Matt
Anthony cc
Anthony
Problem  17
practice icon

Simplify: \(\dfrac{-5(-4)+2(-3)}{2(-1)-5}\)

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Mr. McKeague cc
Mr. McKeague
Ana
Ana
Anthony cc
Anthony
Anthony espanol spanish
Anthony
Problem  18
practice icon

Find the value of the expression \(3x+4y+5\) when \(x\) is \(6\) and \(y\) is \(7\)

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Ana
Ana
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  19

Locate the numbers \(-4.5\), \(-0.75\), \(\dfrac{1}{2}\), \(\sqrt{2}\), \(\pi\), and \(4.1\) on the real number line.

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Mr. Perez
Mr. Perez
Katrina cc
Katrina
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  20
practice icon

Write \(\lvert 5\rvert\) without absolute value symbols.

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Ana
Ana
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  21
practice icon

Write \(\lvert -2\rvert\) without absolute value symbols

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Ana
Ana
Aaron cc
Aaron
Cynthia espanol spanish
Cynthia
Problem  22
practice icon

Write \(-\lvert -3\rvert\) without absolute value symbols

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Ana
Ana
Aaron cc
Aaron
Cynthia espanol spanish
Cynthia
Problem  23
practice icon

Simplify \(5(4x)\) by using the associative property.

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Betsy cc
Betsy
Logan cc
Logan
Cynthia espanol spanish
Cynthia
Problem  24
practice icon

Simplify \(\dfrac{1}{4}(4a)\) by using the associative property.

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Betsy cc
Betsy
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  25
practice icon

Apply the distributive property to \(\dfrac{1}{2}(3x+6)\) and then simplify the result.

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Julieta cc
Julieta
Betsy cc
Betsy
Logan cc
Logan
Julieta cc espanol spanish
Julieta
Problem  26
practice icon

Simplify: \(\,2(3y+4)+2\)

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Betsy cc
Betsy
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  27
practice icon

Apply the distributive property to \(a\left(1+\dfrac{1}{a}\right)\) and then simplify the result.

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Betsy cc
Betsy
Logan cc
Logan
Cynthia espanol spanish
Cynthia
Problem  28
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Apply the distributive property to \(3\left(\dfrac{1}{3}x+5\right)\) and then simplify the result.

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Betsy cc
Betsy
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  29
practice icon

Apply the distributive property to \(6\left(\dfrac{1}{3}x+\dfrac{1}{2}y\right)\) and then simplify the result.

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Betsy cc
Betsy
Logan cc
Logan
Cynthia espanol spanish
Cynthia
Problem  30
practice icon

Use the distributive property to combine similar terms for the expression \(3x+5x\).

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Betsy cc
Betsy
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  31
practice icon

Use the distributive property to combine similar terms for the expression \(3y+y\)

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Betsy cc
Betsy
Logan cc
Logan
Cynthia espanol spanish
Cynthia
Problem  32
practice icon

Simplify: \(7x+4+6x+3\)

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Stefanie cc
Stefanie
Katrina cc
Katrina
Logan cc
Logan
Julieta cc espanol spanish
Julieta
Problem  33
practice icon

Simplify: \(4+3(2y+5)+8y\)

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Stefanie cc
Stefanie
Katrina cc
Katrina
Cynthia espanol spanish
Cynthia
Edwin espanol spanish
Edwin
Problem  34
practice icon

Simplify: \(3(2y-1)+y\)

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Stefanie cc
Stefanie
Katrina cc
Katrina
Keith cc
Keith
Anthony espanol spanish
Anthony
Problem  35
practice icon

Simplify: \(8-3(4x-2)+5x\)

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Mr. McKeague cc
Mr. McKeague
Stefanie cc
Stefanie
Anthony cc
Anthony
David espanol spanish
David
Problem  36
practice icon

Simplify: \(5(2a+3)-(6a-4)\)

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Stefanie cc
Stefanie
Katrina cc
Katrina
Keith cc
Keith
Anthony espanol spanish
Anthony
Problem  37
practice icon

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cup B\).

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Suhas
Suhas
Katrina cc
Katrina
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  38
practice icon

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cap B\).

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Mr. Perez
Mr. Perez
Katrina cc
Katrina
Logan cc
Logan
Julieta cc espanol spanish
Julieta
Problem  39
practice icon

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(A\cap C\).

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Mr. Perez
Mr. Perez
Katrina cc
Katrina
Logan cc
Logan
Edwin espanol spanish
Edwin
Problem  40
practice icon

Let \(A=\{1, 3, 5\}\), \(B=\{0, 2, 4\}\), and \(C=\{1, 2, 3, \ldots\}\). Find \(B\cup C\).

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Mr. Perez
Mr. Perez
Katrina cc
Katrina
Logan cc
Logan
Julieta cc espanol spanish
Julieta
Problem  41
practice icon

For the set \(\left\{-5, \,-3.5, \,0, \,\frac{3}{4}, \,\sqrt{3}, \,\sqrt{5}, \,9\right\}\), list:

  1. whole numbers

  2. integers

  3. rational numbers

  4. irrational numbers

  5. real numbers

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Mr. Perez
Mr. Perez
Katrina cc
Katrina
Graham
Graham
Edwin espanol spanish
Edwin

2
Integer Exponents and Scientific Notation

Problem  1
practice icon

Write each expression using positive exponents.

  1. \(4^{-2}\)

  2. \(1.45^{-3}\)

  3. \(\dfrac{x^{-4}}{y^5}\), \(y\neq 0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Simplify each expression and write it using positive exponents. Assume that variables represent nonzero real numbers.

  1. \(\dfrac{16x^{10}y^4}{4x^{10}y^8}\)

  2. \(\dfrac{\left(3x^{-2}\right)^3}{x^{-5}}\)

  3. \(\left(\dfrac{54s^2t^{-3}}{6zt}\right)^{-2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  3
practice icon

Express each of the following numbers in scientific notation.

  1. \(328.5\)

  2. \(4.69\)

  3. \(0.00712\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4
practice icon

Write the following numbers in decimal form.

  1. \(2.1\times 10^5\)

  2. \(3.47\times 10^{-3}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5
practice icon

In 2012, Japan had a population os \(1.27\times 10^8\) and a land area of \(3.64\times 10^5\) square kilometers. What is the population density of Japan, that is, the number of people per square kilometer of land? Express your answer in scientific notation.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio

3
Roots, Radicals, and Rational Exponents

Problem  1
practice icon

Determine \(\sqrt[3]{64}\), \(\sqrt[4]{64}\), and \(\sqrt[5]{-32}\).

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Rationalize the denominator of each expression.

  1. \(\sqrt{\dfrac{18}{5}}\)

  2. \(\dfrac{\sqrt[3]{10}}{\sqrt[3]{3}}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  3
practice icon

Simplify the following.

  1. \(\sqrt{75}\)

  2. \(\sqrt[3]{\dfrac{125}{108}}\)

  3. \(\sqrt{x^5y^7}\), \(x,y>0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  4
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Rationalize the denominator.
\(\dfrac{5}{4-\sqrt{2}}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5
practice icon

Simplify the following radical expressions. Assume \(x\geq0\).

  1. \(\sqrt{48}+\sqrt{27}-\sqrt{12}\)

  2. \(\left(3+\sqrt{x}\right)\left(4-2\sqrt{x}\right)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  6
practice icon

Evaluate the following.

  1. \(27^\frac{1}{3}\)

  2. \(-36^\frac{1}{2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  7
practice icon

Evaluate the following expressions without using a calculator.

  1. \((4)^\frac{1}{2}\)

  2. \(\left(\dfrac{1}{3}\right)^{-2}\)

  3. \((23.1)^0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  8
practice icon

Simplify and write positive exponents.

  1. \((25)^\frac{1}{2}\)

  2. \(\left(16s^\frac{4}{3}t^{-3}\right)^\frac{3}{2}\), \(s,t>0\)

  3. \(\dfrac{\left(x^{-\frac{2}{3}}y^{-3}\right)^3}{y^{-\frac{1}{2}}x^2}\), \(x,y>0\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Saba cc
Saba

4
Polynomials

Problem  1

Write the following polynomial in descending order and find its degree, terms, coefficients, and constant term. \[-4x^2+3x^5-2x-7\]

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Add or subtract each of the following.

  1. \(\left(3x^3+2x^2-5x+7\right)+\left(x^3-x^2+5x-2\right)\)

  2. \(\left(s^4+\dfrac{3}{4}s^2\right)-\left(s^4-s^2\right)\)

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  3
practice icon

Multiply \(\left(-2x^2\right)\left(4x^7\right)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4
practice icon

Multiply \((3x+2)(-2x-3)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5
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Multiply \((-7x+4)(5x-1)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
practice icon

Multiply \((4y^3-3y+1)(y-2)\)

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  7

Multiply each of the following:

  1. \((3y+4)^2\)

  2. \(\left(4x^2-5\right)^2\)

  3. \((7y+3)(7y-3)\)

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Julieta cc
Julieta
Mr. Galletti cc
Mr. Galletti
Julieta cc espanol spanish
Julieta

5
Factoring

Problem  1

Factor the greatest common factor from each of the following.

  1. \(3x^4+9x^3+18x^2\)

  2. \(-8y^2-6y+4\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Factor \(x^3-x^2+2x-2\) by grouping.

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Mr. Hampton cc
Mr. Hampton
Shelby cc
Shelby
Gordon cc
Gordon
Saba cc
Saba
Problem  3
practice icon

Factor each of the following

  1. \(x^2-2x-8\)

  2. \(8x^3-10x^2-12x\)

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Mr. Hampton cc
Mr. Hampton
Shelby cc
Shelby
Gordon cc
Gordon
Saba cc
Saba
Problem  4

Factor \(6x^3-3x^2-45x\)

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Julieta cc
Julieta
Julieta cc espanol spanish
Julieta
Problem  5

Factor using one of the quadratic factoring patterns.

  1. \(x^2-49\)

  2. \(8x^2+32x+32\)

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Julieta cc
Julieta
Julieta cc espanol spanish
Julieta
Problem  6

Factor using one of the cubic factorization patterns.

  1. \(x^3+27\)

  2. \(27x^3-64\)

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Julieta cc
Julieta
Julieta cc espanol spanish
Julieta

6
Rational Expressions

Problem  1
practice icon

For what value of \(x\) is the following rational expression defined? \[\frac{x+1}{(x-3)(x-5)}\]

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Nathan cc
Nathan
Problem  2
practice icon

Simplify: \(\dfrac{x^2-2x+1}{3x^2-4x+1}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  3

Multiply the following and express your answer in lowest terms. For what values of the variable is the expression meaningful?

  1. \(\dfrac{3a}{8}\cdot\dfrac{24}{6a^3}\)

  2. \(\dfrac{x^2+x-6}{x^2-4}\cdot\dfrac{(x+2)^2}{\left(x^2+9\right)}\)

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Saba cc
Saba
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  4
practice icon

Divide and simplify:
\(\dfrac{3x^2-5x-2}{x^2-4x+4}\div\dfrac{9x^2-1}{x+5}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  5

Add or subtract the following and express your answer in lowest terms.

  1. \(\dfrac{x+4}{3x+6}+\dfrac{2x+1}{x^2+7x+10}\)

  2. \(\dfrac{3x}{4-2x}-\dfrac{x+5}{x^2-4}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
practice icon

Simplify \(\dfrac{\dfrac{x^2-4}{2x+1}}{\dfrac{x^2+x-6}{x-1}}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  7
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Simplify \(\dfrac{\dfrac{1}{x}+\dfrac{1}{xy}}{\dfrac{3}{y^2}+\dfrac{1}{y}}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta
Julieta