algebra 2 module 1 solving equations assignment answer key

Infinite Algebra 2

Test and worksheet generator for algebra 2.

Infinite Algebra 2 covers all typical Algebra 2 material, beginning with a few major Algebra 1 concepts and going through trigonometry. There are over 125 topics in all, from multi-step equations to trigonometric identities. Suitable for any class with advanced algebra content. Designed for all levels of learners, from remedial to advanced.

Included in version 2.70.06 released 1/5/2023:

• Fixed: Choices could appear incorrectly

Included in version 2.70.05 released 12/21/2022:

• Fixed: Continuous Exponential Growth and Decay Word Problems - Could generate an incorrect answer

Included in version 2.70 released 7/8/2022:

• New: Add preferences for Metric/Imperial units
• New: Add preference to “Prefer x“ as the variable letter
• New topic: Discrete Relations
• New topic: Continuous Relations
• New topic: Evaluating and Graphing Functions
• New topic: Direct and Inverse Variation
• New topic: Continuous Exponential Growth and Decay
• Improved: UI, security, and stability with updated libraries
• Fixed: Properties of Exponents – Choices could be equivalent
• Fixed: Properties of Logarithms – Choices could be equivalent
• Fixed: Solving Exponential Equations With Logarithms – Choices could be equivalent
• Fixed: Geometric Series – Choices could be equivalent

Included in version 2.62 released 2/8/2022:

• New: Print questions from Presentation View
• New Topic: Discrete Exponential Growth and Decay Word Problems
• Improved: UI, security, and stability with updated framework and libraries
• Improved: [Mac] Dark mode
• Fixed: [Windows] - Detached topic list does not close when application is closed
• Fixed: Probability Mutually Exclusive Word Problems - mutually exclusive case given when inappropriate

Included in version 2.61.03 released 9/10/2021:

• Improved: Minor UI improvements
• Fixed: Law of Cosines - Crash when opening topic
• Fixed: [Windows] Unable to change page layout to landscape
• Fixed: [Windows] Error 'Unable to load platform plugin'

Included in version 2.61 released 8/27/2021:

• Fixed: Preferences page for Kuta Works
• Improved: Security and stability with updated networking libraries

Included in version 2.60 released 8/19/2021:

• Improved: Site licenses check expiration date more frequently
• Fixed: Combinatoric Word Problems - Values could overflow
• Fixed: Properties of Exponents - Answer might not be fully simplified
• Fixed: Properties of Logs - Possible to have duplicate choices
• Fixed: Radical Equations - Option to mix radicals and rational exponents had no effect

Included in version 2.52 released 6/14/2019:

• New: Scramble questions by directions
• New: Scramble all questions in assignment
• New: Consolidate question sets with identical options
• New: Regroup question sets by directions
• New: Kuta Works - Create a two semester course
• New: Kuta Works - Clone assignments from a previous course into a new one
• Fixed: Presentation View - Arrow keys could change zoom at start/end of assignment
• Fixed: Tight Layout - Directions could be cramped for no reason
• Fixed: Inverse Trig - Angles limited for tan( x )
• Fixed: Inverse Trig - Errors in UI logic

Included in version 2.50 released 4/12/2019:

• New: Kuta Works - Option to hide answers and results from students until after due date
• New: Kuta Works - Option to control how long choices are hidden
• Improved: Options windows appear at a better initial size
• Improved: Window controls shown at their native size and spacing
• Improved: Network proxy configuration window
• Improved: License activation process
• Fixed: Presentation View - Answers could be cut off
• Fixed: Some window controls cut off on Windows 10 with display scaling
• Fixed: Spelling errors
• Fixed: Kuta Works - Course list could display more than just your active courses
• Fixed: Graphing Rational Functions - Open holes in graph could appear as filled holes

Included in version 2.42 released 12/11/2018:

• Fixed: Certain symbols incorrect if saved on Mac and loaded on Windows or vice versa
• Fixed: General Sequences - Recursive formula sometimes wrong
• Fixed: Properties of Logarithms - Avoid questions with reducible roots

Included in version 2.41 released 9/26/2018:

• Fixed: Archived courses in Kuta Works are now hidden
• Fixed: Generating or regenerating questions could cause a crash

Included in version 2.40 released 8/8/2018:

• New: Integrated with Kuta Works . Now post assignments online.

Included in version 2.25 released 4/27/2018:

• Improved: Generate questions more quickly
• Improved: Measurement arrows reach correct endpoints
• Improved: Absolute Value Equations - Better decimal numbers
• Improved: Absolute Value Inequalities - Better decimal numbers
• Improved: Compound Inequalities - Better decimal numbers
• Improved: Multi-Step Equations - Better decimal numbers and more predictable special cases
• Improved: Multi-Step Inequalities - Better decimal numbers
• Improved: Order of Operations - Better decimal numbers
• Fixed: Trig units for piecewise functions and ordered pairs
• Fixed: Graphing Rational Functions - Prevent equivalent choices
• Fixed: Law of Sines - Program could freeze
• Fixed: Transformations - Prevent equivalent choices

Included in version 2.18 released 7/31/2017:

• Fixed: Open circles on graphs were filled in when displayed as answer in red
• Fixed: Typo in sample custom question
• Fixed: Transformations - Program could freeze
• Fixed: Systems of Quadratic Equations - Program could freeze

Included in version 2.17 released 4/27/2017:

• Improved: [Windows] "Export to clipboard as bitmap" renders image with better quality
• Fixed: Probability with Combinations and Permutations - Could ask about impossible lottery ball

Included in version 2.16.20 released 11/16/2016:

• Fixed: Solving Quadratic Equations by Taking Square Roots - Option to "Allow fractions" not working as expected

Included in version 2.16 released 9/16/2016:

• Fixed: Numbers with many significant digits could round incorrectly
• Fixed: Customized question containing an error could not be modified
• Fixed: Customized question containing an error could cause a crash
• Fixed: Feedback tool detects Windows 10 properly
• Fixed: Mouse cursor when drag-dropping a question
• Fixed: Keyboard shortcut for changing order of question sets in assignment

Included in version 2.15 released 7/14/2016:

• Improved: Reorder question sets by dragging and dropping
• Improved: Move individual questions within a set by dragging and dropping
• Improved: Presentation View - Commands now on a movable, dockable toolbar
• Improved: Added option for large toolbar icons
• Improved: New icons
• Fixed: Certain options could result in double parenthesis or parenthesis around a negative sign
• Fixed: Arrow over vector variable name not red when appropriate
• Fixed: Solving Systems of Equations by Graphing - Answer could be displayed twice
• Fixed: Equations of Parabolas - Certain givens indicate wrong sign

Included in version 2.11 released 5/13/2016:

• Improved: [Windows] "Export to clipboard as bitmap" pastes into Microsoft Word at appropriate DPI
• Fixed: Keyboard shortcut for adding space to question set

Included in version 2.10 released 5/11/2016:

• New: Preference for including "None of the above" as a choice
• New: One column layout
• New: Added support for Infinite Precalculus
• New: Easily add piecewise functions of graphs in custom questions: Example: piecewise([2x-3] if [x<5], [x-1] if [x >= 5])
• New: Easily add functions with restricted domains to graphs in custom questions Example: function(x/2, x<0)
• Improved: More efficient layout of choices
• Improved: [Mac] Added option for legal paper in page setup
• Fixed: Spelling and grammatical errors
• Fixed: Graphing Quadratic Functions - Prevent equivalent choices
• Fixed: Binomial Theorem - Certain options didn't allow for many questions
• Fixed: Right Triangle Trig - Couldn't select just one ratio except sine
• Fixed: Right Triangle Trig - Answer sometimes given in problem
• Fixed: Law of Cosines - Certain SAS cases could freeze program
• Note: Older versions will not be able to open assignments saved with this version

Included in version 2.06 released 8/3/2015:

• Fixed: Crash when saving if previously used folder no longer exists

Included in version 2.05 released 7/20/2015:

• New Topic: Sample Spaces and The Fundamental Counting Principle
• New Topic: Independent and Dependent Events
• New Topic: Independent and Dependent Events Word Problems
• New Topic: Mutually Exclusive Events
• New Topic: Mutually Exclusive Events Word Problems
• Fixed: Classifying Conics - Checkboxes for conic types had no effect
• Fixed: Presentation View - Displayed wrong directions in 3-up mode
• Fixed: Presentation View - Lines too long after creating new problems
• Fixed: [Windows] Uncommon error when using the network

Included in version 2.04.40 released 3/31/2015:

• Fixed: Activation data unrecognized under special circumstances
• Fixed: [Windows] Help files broken by previous release
• Fixed: Order of Operations - Radio buttons reversed

Included in version 2.04.20 released 3/23/2015:

• Improved: [Windows] "Export to clipboard as bitmap" uses a transparent background

Included in version 2.04 released 3/17/2015:

• New Topic: Permutations
• New Topic: Combinations
• New Topic: Permutations vs Combinations
• New Topic: Probability with Permutations and Combinations
• New Topic: Converting Degrees and Degrees-Minutes-Seconds
• [factorial(n)]
• [perm(n, r)]
• [comb(n, r)]
• Improved: Better support for proxies
• Improved: Support for HiDPI Mac Retina displays
• Improved: Print options window - Some settings are now persistent
• Improved: Presentation View - Use arrow keys to navigate between questions
• Improved: Question set updates free response/multiple-choice when all questions have changed
• Improved: [Mac] Activation also checks /Library/Preferences/ks-config.txt for serial numbers
• Fixed: [Windows] Crash when printing with certain printers
• Fixed: Crash when loading an assignment with over 80 custom questions
• Fixed: Custom question answer was never choice A
• Fixed: Paper size set before printing
• Fixed: Cramer's Rule - Fixed directions
• Fixed: Systems of Three Equations - Restored the option to "only solve for x"
• Fixed: Center of circle wasn't appearing

Included in version 2.03 released 9/19/2014:

• Fixed: Preference for question layout mode not correctly loaded
• Fixed: Issues with automatic spacing and highlighting questions when using tight layout
• Fixed: In Presentation View, graphing problems display choices more compactly
• Fixed: Printing blank pages on HP LaserJet 1018, 1020, and 1022 among others
• Fixed: Logs and Exponents as Inverses - problem with slider

Included in version 2.02 released 8/5/2014:

• New: Software uses system proxy when necessary
• Improved: Activation window has taskbar entry
• Improved: Remembers the last directory used when saving and opening files
• Fixed: [Windows only] Possible crash when loading an assignment with a lot of custom questions
• Fixed: [Mac only] Application stops responding to hover events
• Fixed: Activation from before v2.0 sometimes not recognized
• Fixed: Export to clipboard sometimes cuts off choices B and D
• Fixed: Help contents could become unusable
• Fixed: Word wrap when modifying a question
• Fixed: Memory leak

Included in version 2.01 released 7/28/2014:

• Fixed: Could crash when opening an assignment from a different program

Included in version 2.00 released 7/23/2014:

• New: Print to PDF
• New: Filter for topic list
• New: Zoom while viewing on screen
• New: "More like these" button in presentation mode
• New: Show where punch holes will go on the page
• New: Print only an answer sheet
• New: When printing, page range is enabled
• New: Page elements are outlined when hovered
• New: Double-click on directions to change them
• New: When merging assignments, options to put similar topics together or put them end-to-end
• New: Highlight and go to the questions in a question set (menu command or shift-click)
• New: Site licenses can be activated per-machine if run by the administrator
• Ctrl-Click on a question
• Shift-Click on a question
• Ctrl-Shift-Click on a question
• Shift-Click in a empty area of a page
• Ctrl-Shift-Click in a empty area of a page
• Shift-Click on a question set
• Ctrl-Shift-Click on a question set
• Improved: Assignment files are drastically smaller
• Improved: Sidebar is dockable on either side, can be floated, and position reset
• Improved: Auto-spacing doesn't leave dead space at bottom of each page
• Improved: Preview when editing directions
• Improved: Keyboard shortcuts work no matter where the focus is
• Improved: Scale number of questions window more user-friendly
• Fixed: Minor bugs in seven topics

Included in version 1.56 released 8/14/2013:

• Fixed: Graphing Exponential Functions - Correct choice used lighter shade of blue
• Fixed: Radical Equations could be missing a solution
• Fixed: Graphs of Polynomial Functions - Inflection points sometimes listed as local extrema
• Fixed: Properties of Exponents - Occasionally answer wasn't fully simplified
• Fixed: Dividing Polynomials - Occasionally polynomial was evenly divisible when it shouldn't have been
• Fixed: Graphing Rational Functions - Minor UI issue
• Fixed: Geometric Sequences - Given information sometimes allowed for two possible common ratios
• Fixed: Dividing Polynomials could freeze
• Fixed: Systems of Equations by Elimination - "No solution" occasionally should have been "Infinite solutions"

Included in version 1.55 released 12/11/2012:

• New: Graphs can be added to custom questions
• Improved: Graphing and Graph Paper utility more powerful and easier to use
• Improved: Support for loading files from Infinite Calculus
• Improved: Faster save/load
• Fixed: Answer for Factoring Quadratic Expressions sometimes incorrect
• Fixed: Choices for Evaluating Functions with a variable operand could contain wrong variable
• Fixed: Choices for Function Operations with a variable operand could contain wrong variable
• Fixed: Function Operations, functions when composed with self could provide the wrong answer
• Fixed: Custom questions with an illegal expression could freeze the program
• Fixed: Certain families of functions graphed incorrectly

Included in version 1.53 released 9/11/2012:

• New: Preference for notation for greatest integer function
• New: Added maximize/restore button to Presentation View
• New: Graphing Absolute Value Equations has options for including coefficients
• Improved: Help files
• Improved: Scroll bars
• Improved: User interface
• Improved: When choices make a question too tall for a page, some choices are removed
• Improved: Algebraic simplification routines are now more efficient
• Improved: Better graphs for: Graphing Absolute Value Equations, Graphing Linear Equations, Graphing Exponential Functions, Graphing Quadratic Functions, Graphing Linear Inequalities, Graphing Systems of Linear Inequalities
• Improved: Better number lines for: Multi-Step Inequalities, Compound Inequalities, Absolute Value Inequalities
• Fixed: Wrapping to full-page and half-page could be too wide
• Fixed: Able to graph certain families of functions
• Fixed: Issue involving loading & regenerating an assignment and answers being hidden
• Fixed: Program could ask about changing directions too much
• Fixed: Graphs for Compound Inequalities could omit interval
• Fixed: Double & Half Angle Identities: Button to show more examples did not work
• Fixed: Writing Linear Equations: Answer can't be line given in question

Included in version 1.52 released 5/29/2012:

• Improved: More professional radical signs
• Improved: Graphs of discontinuous functions have breaks and open / closed holes
• Improved: Even less likely to crash when generating questions
• Fixed: Word wrapping could skip blank lines, not wrap where appropriate
• Fixed: Wrapping to full page in a custom question was too wide
• Fixed: Products of powers of e (like 2 e &sup3;) no longer display with a multiplication dot
• Fixed: Most diagrams appeared incorrectly in print preview
• Fixed: Random error message on save or load
• Fixed: Graphing Rational Functions: Open holes in graph could appear as filled holes

Included in version 1.51.02 released on 4/9/2012:

• Fixed: Minimum Windows version was set incorrectly

Included in version 1.51 released 4/5/2012:

• New: Added greatest integer function / floor function to custom questions    Example: Find [int(2x)] when [x=3/5]
• New: Added piecewise functions to custom questions    Example: [eval(f,x)] = [piecewise [x] if [x < 0], [xx] if [x >= 0]]
• Improved: Faster! Optimized rendering of questions to screen
• Improved: Faster! Optimized graphing routines
• Improved: Improved graphing capabilities
• Improved: Diagrams drawn more smoothly on screen
• Fixed: Rewriting Logarithms - Numeric problems sometimes had bad values
• Fixed: Evaluating Logarithms - Problems sometimes had bad values
• Fixed: Graphs could omit holes
• Fixed: Graphs of constant functions or those involving e and π could be incorrect
• Fixed: Minor indentation issue in custom questions
• Fixed: Horizontal asymptotes could be drawn beyond a graph's area

Included in version 1.50 released 3/15/2012:

• New: Presentation View window is resizeable
• New: Presentation View has option to automatically hide the answers when a new question is displayed
• New: High-level filter to prevent questions from containing an illegal expression
• Improved: Function notation can now be edited
• Improved: Faster! Optimized the simplification of mathematical expressions
• Improved: Faster! Improved undo/redo algorithm
• Improved: Smaller executable size
• Improved: "Current Question Sets" list easier to use
• Improved: Options windows resize more smoothly
• Fixed: Equation of Parabolas could freeze
• Fixed: Geometric Series questions could list "Illegal Expression" as a choice
• Fixed: Graphing Exponential Functions could freeze
• Fixed: Systems of Equations Word Problems could freeze
• Fixed: Punctuation in some word problems
• Fixed: Certain expressions in a custom question would cause the software to crash
• Fixed: Expressions like root × term would not print a multiplication dot
• Note: Beginning with this version, Windows XP SP3 is the minimum required version of Windows

Included in version 1.45 released 4/12/2011:

• New: Assignments from this program can be opened and modified by our other programs          (Assignments saved with v1.45 or greater can be opened by other programs v1.45 or greater)
• New: Student data fields (name, date, period) can be renamed
• New: Additional student data field available
• New: Add & Continue is available when modifying an existing question set
• New: Link to this details page when a software update is available
• New: Product serial numbers can be placed in config.txt to facilitate enterprise installations
• Improved: Better backwards- and future-compatibility
• Fixed: Assignments with quotes in the title don't prevent Save As
• Fixed: Quotients could be sometimes be simplified incorrectly
• Fixed: Choices for dilations sometimes inappropriate
• Fixed: Dilations rounding x- and y-coordinates
• Fixed: Function Operations could have "Illegal Expression" as an answer or as a choice
• Fixed: Multi-Step Equations: Program could freeze
• Fixed: Dividing Radical Expressions: Program could freeze
• Fixed: Add & Continue respects change in problem type
• Fixed: Powers of i are correctly displayed

Included in version 1.42 released on 6/28/2010:

• New: Automatically checks for updates
• New: List topics by index order or suggested order

Included in version 1.40 released 2/8/2010:

• New Preference: 'Spacious' or 'tight' layout of questions on page
• Improved: Question sets with one question are more intelligently spaced
• Improved: Graphs with small ranges now look better
• Improved: Options screen now resizeable
• Fixed: Directions occasionally orphaned at bottom of page
• Fixed: Correct numbers changed because zeros at the end of a number were deleted (50 --> 5)

Included in version 1.08.20 released 5/22/2009:

• Changed: Minor change to license agreement so that renewals extend the termination date instead of beginning a new term.

Included in version 1.08 released 4/10/2009:

• New: Possible to change directions for a set without regenerating questions
• Improved: Changing directions cascades changes to appropriate neighbors
• Correction: Answers to Properties of Exponents could have wrong sign
• Correction: Inequalities were sometimes solved incorrectly
• Fixed: Software sometimes failed to renumber questions in assignment

Included in version 1.07 released 1/6/2009:

• New: Trig. equations that don't require factoring
• Correction: Function operations / linear combinations answers sometimes wrong
• Fixed: Software crashes when starting if closed when side panel is minimized

Included in version 1.06 released 12/4/2008:

• Improved: Isolate up to four questions on screen at once
• Improved: Support for computers with multiple processors
• Improved: User interface: Toolbar + Preferences
• Fixed: Obscure error involving selecting a question and scaling the assignment

Included in version 1.03 released 10/1/2008:

• New: Export individual questions as bitmaps
• Improved: Custom questions wrap to whole/half page
• Improved: Insert special symbols and math in custom questions
• Improved: Insert 'blanks' into math text
• Improved: Line thickness preference also now controls darkness of grid lines in graphs
• Correction: Writing logs in terms of others sometimes created duplicate questions
• Fixed: Internet calls would sometimes report wrong error

Included in version 1.02 released 9/18/2008:

• Correction: Question type was not saving all of its options
• Correction: Equations of Hyperbolas description was sometimes wrong
• Correction: Geometric Series directions was sometimes wrong
• Correction: Binomial Theorem type could cause the program to freeze
• Correction: Quadratic/Linear systems sometimes had the wrong answer
• Correction: Rational Exponents answers now correct
• Fixed: Internet calls could freeze program

Try Infinite Algebra 2 Today!

Discover the power and flexibility of our software firsthand with a free, 14-day trial. Installation is fast and simple. Within minutes, you can have the software installed and create the precise worksheets you need -- even for today's lesson.

Use each trial for up to 14 days. The trial version is identical to the retail version except that you cannot print to electronic formats such as PDF.

Algebra 2 Worksheets with answer keys

Enjoy these free printable math worksheets . Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Absolute Value Equations
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (advanced)
• End of Unit, Review Sheet
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (no logs)
• Compound Interest Worksheet (logarithms required)
• Simplify Rational Exponents
• Solve Equations with Rational Exponents
• Solve Equations with variables in Exponents
• Factor by Grouping
• 1 to 1 functions
• Evaluating Functions
• Composition of Functions
• Inverse Functions
• Operations with Functions
• Functions Review Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Logarithmic Equations Worksheet
• Dividing Polynomials Worksheet
• Solve Quadratic Equations by Factoring
• Solve Quadratic Equations by Completing the Square
• Quadratic formula Worksheet (real solutions)
• Quadratic Formula Worksheet (complex solutions)
• Quadratic Formula Worksheet (both real and complex solutions)
• Discriminant Worksheet
• Sum and Product of Roots
• Radical Equations
• Rationalizing the Denominator
• Simplify Rational Expressions Worksheet
• Dividing Rational Expressions
• Multiplying Rational Expressions
• Adding and Subtracting Rational Expressions (with like denominators)
• Adding and Subtracting Ratioal Expressions with Unlike Denominators
• Mixed Review on Rational Expressions

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

2.1 The Rectangular Coordinate Systems and Graphs

x -intercept is ( 4 , 0 ) ; ( 4 , 0 ) ; y- intercept is ( 0 , 3 ) . ( 0 , 3 ) .

125 = 5 5 125 = 5 5

( − 5 , 5 2 ) ( − 5 , 5 2 )

2.2 Linear Equations in One Variable

x = −5 x = −5

x = −3 x = −3

x = 10 3 x = 10 3

x = 1 x = 1

x = − 7 17 . x = − 7 17 . Excluded values are x = − 1 2 x = − 1 2 and x = − 1 3 . x = − 1 3 .

x = 1 3 x = 1 3

m = − 2 3 m = − 2 3

y = 4 x −3 y = 4 x −3

x + 3 y = 2 x + 3 y = 2

Horizontal line: y = 2 y = 2

Parallel lines: equations are written in slope-intercept form.

y = 5 x + 3 y = 5 x + 3

2.3 Models and Applications

C = 2.5 x + 3 , 650 C = 2.5 x + 3 , 650

L = 37 L = 37 cm, W = 18 W = 18 cm

2.4 Complex Numbers

−24 = 0 + 2 i 6 −24 = 0 + 2 i 6

( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i ( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i

5 2 − i 5 2 − i

18 + i 18 + i

−3 −4 i −3 −4 i

2.5 Quadratic Equations

( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1 ( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1

( x −7 ) ( x + 3 ) = 0 , ( x −7 ) ( x + 3 ) = 0 , x = 7 , x = 7 , x = −3. x = −3.

( x + 5 ) ( x −5 ) = 0 , ( x + 5 ) ( x −5 ) = 0 , x = −5 , x = −5 , x = 5. x = 5.

( 3 x + 2 ) ( 4 x + 1 ) = 0 , ( 3 x + 2 ) ( 4 x + 1 ) = 0 , x = − 2 3 , x = − 2 3 , x = − 1 4 x = − 1 4

x = 0 , x = −10 , x = −1 x = 0 , x = −10 , x = −1

x = 4 ± 5 x = 4 ± 5

x = 3 ± 22 x = 3 ± 22

x = − 2 3 , x = − 2 3 , x = 1 3 x = 1 3

2.6 Other Types of Equations

{ −1 } { −1 }

0 , 0 , 1 2 , 1 2 , − 1 2 − 1 2

1 ; 1 ; extraneous solution − 2 9 − 2 9

−2 ; −2 ; extraneous solution −1 −1

−1 , −1 , 3 2 3 2

−3 , 3 , − i , i −3 , 3 , − i , i

2 , 12 2 , 12

−1 , −1 , 0 0 is not a solution.

2.7 Linear Inequalities and Absolute Value Inequalities

[ −3 , 5 ] [ −3 , 5 ]

( − ∞ , −2 ) ∪ [ 3 , ∞ ) ( − ∞ , −2 ) ∪ [ 3 , ∞ )

x < 1 x < 1

x ≥ −5 x ≥ −5

( 2 , ∞ ) ( 2 , ∞ )

[ − 3 14 , ∞ ) [ − 3 14 , ∞ )

6 < x ≤ 9 ​ or ( 6 , 9 ] 6 < x ≤ 9 ​ or ( 6 , 9 ]

( − 1 8 , 1 2 ) ( − 1 8 , 1 2 )

| x −2 | ≤ 3 | x −2 | ≤ 3

k ≤ 1 k ≤ 1 or k ≥ 7 ; k ≥ 7 ; in interval notation, this would be ( − ∞ , 1 ] ∪ [ 7 , ∞ ) . ( − ∞ , 1 ] ∪ [ 7 , ∞ ) .

2.1 Section Exercises

Answers may vary. Yes. It is possible for a point to be on the x -axis or on the y -axis and therefore is considered to NOT be in one of the quadrants.

The y -intercept is the point where the graph crosses the y -axis.

The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , 6 ) . ( 0 , 6 ) .

The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , −3 ) . ( 0 , −3 ) .

The x- intercept is ( 3 , 0 ) ( 3 , 0 ) and the y -intercept is ( 0 , 9 8 ) . ( 0 , 9 8 ) .

y = 4 − 2 x y = 4 − 2 x

y = 5 − 2 x 3 y = 5 − 2 x 3

y = 2 x − 4 5 y = 2 x − 4 5

d = 74 d = 74

d = 36 = 6 d = 36 = 6

d ≈ 62.97 d ≈ 62.97

( 3 , − 3 2 ) ( 3 , − 3 2 )

( 2 , −1 ) ( 2 , −1 )

( 0 , 0 ) ( 0 , 0 )

y = 0 y = 0

not collinear

A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 ) A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 )

d = 8.246 d = 8.246

d = 5 d = 5

( −3 , 4 ) ( −3 , 4 )

x = 0          y = −2 x = 0          y = −2

x = 0.75 y = 0 x = 0.75 y = 0

x = − 1.667 y = 0 x = − 1.667 y = 0

15 − 11.2 = 3.8 mi 15 − 11.2 = 3.8 mi shorter

6 .0 42 6 .0 42

Midpoint of each diagonal is the same point ( 2 , –2 ) ( 2 , –2 ) . Note this is a characteristic of rectangles, but not other quadrilaterals.

2.2 Section Exercises

It means they have the same slope.

The exponent of the x x variable is 1. It is called a first-degree equation.

If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).

x = 2 x = 2

x = 2 7 x = 2 7

x = 6 x = 6

x = 3 x = 3

x = −14 x = −14

x ≠ −4 ; x ≠ −4 ; x = −3 x = −3

x ≠ 1 ; x ≠ 1 ; when we solve this we get x = 1 , x = 1 , which is excluded, therefore NO solution

x ≠ 0 ; x ≠ 0 ; x = − 5 2 x = − 5 2

y = − 4 5 x + 14 5 y = − 4 5 x + 14 5

y = − 3 4 x + 2 y = − 3 4 x + 2

y = 1 2 x + 5 2 y = 1 2 x + 5 2

y = −3 x − 5 y = −3 x − 5

y = 7 y = 7

y = −4 y = −4

8 x + 5 y = 7 8 x + 5 y = 7

Perpendicular

m = − 9 7 m = − 9 7

m = 3 2 m = 3 2

m 1 = − 1 3 ,   m 2 = 3 ;   Perpendicular . m 1 = − 1 3 ,   m 2 = 3 ;   Perpendicular .

y = 0.245 x − 45.662. y = 0.245 x − 45.662. Answers may vary. y min = −50 , y max = −40 y min = −50 , y max = −40

y = − 2.333 x + 6.667. y = − 2.333 x + 6.667. Answers may vary. y min = −10 ,   y max = 10 y min = −10 ,   y max = 10

y = − A B x + C B y = − A B x + C B

The slope for  ( −1 , 1 ) to  ( 0 , 4 ) is  3. The slope for  ( −1 , 1 ) to  ( 2 , 0 ) is  − 1 3 . The slope for  ( 2 , 0 ) to  ( 3 , 3 ) is  3. The slope for  ( 0 , 4 ) to  ( 3 , 3 ) is  − 1 3 . The slope for  ( −1 , 1 ) to  ( 0 , 4 ) is  3. The slope for  ( −1 , 1 ) to  ( 2 , 0 ) is  − 1 3 . The slope for  ( 2 , 0 ) to  ( 3 , 3 ) is  3. The slope for  ( 0 , 4 ) to  ( 3 , 3 ) is  − 1 3 .

Yes they are perpendicular.

2.3 Section Exercises

Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading.

2 , 000 − x 2 , 000 − x

v + 10 v + 10

Ann: 23 ; 23 ; Beth: 46 46

20 + 0.05 m 20 + 0.05 m

90 + 40 P 90 + 40 P

50 , 000 − x 50 , 000 − x

She traveled for 2 h at 20 mi/h, or 40 miles.

$5,000 at 8% and $15,000 at 12%

B = 100 + .05 x B = 100 + .05 x

R = 9 R = 9

r = 4 5 r = 4 5 or 0.8

W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14 W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14

f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21 f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21

m = − 5 4 m = − 5 4

h = 2 A b 1 + b 2 h = 2 A b 1 + b 2

length = 360 ft; width = 160 ft

A = 88 in . 2 A = 88 in . 2

h = V π r 2 h = V π r 2

r = V π h r = V π h

C = 12 π C = 12 π

2.4 Section Exercises

Add the real parts together and the imaginary parts together.

Possible answer: i i times i i equals -1, which is not imaginary.

−8 + 2 i −8 + 2 i

14 + 7 i 14 + 7 i

− 23 29 + 15 29 i − 23 29 + 15 29 i

8 − i 8 − i

−11 + 4 i −11 + 4 i

2 −5 i 2 −5 i

6 + 15 i 6 + 15 i

−16 + 32 i −16 + 32 i

−4 −7 i −4 −7 i

2 − 2 3 i 2 − 2 3 i

4 − 6 i 4 − 6 i

2 5 + 11 5 i 2 5 + 11 5 i

1 + i 3 1 + i 3

( 3 2 + 1 2 i ) 6 = −1 ( 3 2 + 1 2 i ) 6 = −1

5 −5 i 5 −5 i

9 2 − 9 2 i 9 2 − 9 2 i

2.5 Section Exercises

It is a second-degree equation (the highest variable exponent is 2).

We want to take advantage of the zero property of multiplication in the fact that if a ⋅ b = 0 a ⋅ b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0 o r b = 0. a = 0 o r b = 0.

One, when no linear term is present (no x term), such as x 2 = 16. x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d . ( a x + b ) 2 = d .

x = 6 , x = 6 , x = 3 x = 3

x = − 5 2 , x = − 5 2 , x = − 1 3 x = − 1 3

x = 5 , x = 5 , x = −5 x = −5

x = − 3 2 , x = − 3 2 , x = 3 2 x = 3 2

x = −2 , 3 x = −2 , 3

x = 0 , x = 0 , x = − 3 7 x = − 3 7

x = −6 , x = −6 , x = 6 x = 6

x = 6 , x = 6 , x = −4 x = −4

x = 1 , x = 1 , x = −2 x = −2

x = −2 , x = −2 , x = 11 x = 11

z = 2 3 , z = 2 3 , z = − 1 2 z = − 1 2

x = 3 ± 17 4 x = 3 ± 17 4

One rational

Two real; rational

x = − 1 ± 17 2 x = − 1 ± 17 2

x = 5 ± 13 6 x = 5 ± 13 6

x = − 1 ± 17 8 x = − 1 ± 17 8

x ≈ 0.131 x ≈ 0.131 and x ≈ 2.535 x ≈ 2.535

x ≈ − 6.7 x ≈ − 6.7 and x ≈ 1.7 x ≈ 1.7

a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a

x ( x + 10 ) = 119 ; x ( x + 10 ) = 119 ; 7 ft. and 17 ft.

maximum at x = 70 x = 70

The quadratic equation would be ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. The two values of x x are 20 and 60.

2.6 Section Exercises

This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.

He or she is probably trying to enter negative 9, but taking the square root of −9 −9 is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in −27. −27.

A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.

x = 81 x = 81

x = 17 x = 17

x = 8 ,     x = 27 x = 8 ,     x = 27

x = −2 , 1 , −1 x = −2 , 1 , −1

y = 0 ,     3 2 ,     − 3 2 y = 0 ,     3 2 ,     − 3 2

m = 1 , −1 m = 1 , −1

x = 2 5 , ±3 i x = 2 5 , ±3 i

x = 32 x = 32

t = 44 3 t = 44 3

x = −2 x = −2

x = 4 , −4 3 x = 4 , −4 3

x = − 5 4 , 7 4 x = − 5 4 , 7 4

x = 3 , −2 x = 3 , −2

x = 1 , −1 , 3 , -3 x = 1 , −1 , 3 , -3

x = 2 , −2 x = 2 , −2

x = 1 , 5 x = 1 , 5

x ≥ 0 x ≥ 0

x = 4 , 6 , −6 , −8 x = 4 , 6 , −6 , −8

2.7 Section Exercises

When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.

( − ∞ , ∞ ) ( − ∞ , ∞ )

We start by finding the x -intercept, or where the function = 0. Once we have that point, which is ( 3 , 0 ) , ( 3 , 0 ) , we graph to the right the straight line graph y = x −3 , y = x −3 , and then when we draw it to the left we plot positive y values, taking the absolute value of them.

( − ∞ , 3 4 ] ( − ∞ , 3 4 ]

[ − 13 2 , ∞ ) [ − 13 2 , ∞ )

( − ∞ , 3 ) ( − ∞ , 3 )

( − ∞ , − 37 3 ] ( − ∞ , − 37 3 ]

All real numbers ( − ∞ , ∞ ) ( − ∞ , ∞ )

( − ∞ , − 10 3 ) ∪ ( 4 , ∞ ) ( − ∞ , − 10 3 ) ∪ ( 4 , ∞ )

( − ∞ , −4 ] ∪ [ 8 , + ∞ ) ( − ∞ , −4 ] ∪ [ 8 , + ∞ )

No solution

( −5 , 11 ) ( −5 , 11 )

[ 6 , 12 ] [ 6 , 12 ]

[ −10 , 12 ] [ −10 , 12 ]

x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 ,   ( − 2 , + ∞ ) x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 ,   ( − 2 , + ∞ )

x < − 3   or   x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ ​ ​ [ 1 , ∞ ) x < − 3   or   x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ ​ ​ [ 1 , ∞ )

( − ∞ , −1 ) ∪ ( 3 , ∞ ) ( − ∞ , −1 ) ∪ ( 3 , ∞ )

[ −11 , −3 ] [ −11 , −3 ]

It is never less than zero. No solution.

Where the blue line is above the orange line; point of intersection is x = − 3. x = − 3.

( − ∞ , −3 ) ( − ∞ , −3 )

Where the blue line is above the orange line; always. All real numbers.

( − ∞ , − ∞ ) ( − ∞ , − ∞ )

( −1 , 3 ) ( −1 , 3 )

( − ∞ , 4 ) ( − ∞ , 4 )

{ x | x < 6 } { x | x < 6 }

{ x | −3 ≤ x < 5 } { x | −3 ≤ x < 5 }

( −2 , 1 ] ( −2 , 1 ]

( − ∞ , 4 ] ( − ∞ , 4 ]

Where the blue is below the orange; always. All real numbers. ( − ∞ , + ∞ ) . ( − ∞ , + ∞ ) .

Where the blue is below the orange; ( 1 , 7 ) . ( 1 , 7 ) .

x = 2 , − 4 5 x = 2 , − 4 5

( −7 , 5 ] ( −7 , 5 ]

80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400 80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400

[ 1 , 600 , 2 , 400 ] [ 1 , 600 , 2 , 400 ]

Review Exercises

x -intercept: ( 3 , 0 ) ; ( 3 , 0 ) ; y -intercept: ( 0 , −4 ) ( 0 , −4 )

y = 5 3 x + 4 y = 5 3 x + 4

72 = 6 2 72 = 6 2

620.097 620.097

midpoint is ( 2 , 23 2 ) ( 2 , 23 2 )

x = 4 x = 4

x = 12 7 x = 12 7

y = 1 6 x + 4 3 y = 1 6 x + 4 3

y = 2 3 x + 6 y = 2 3 x + 6

females 17, males 56

x = − 3 4 ± i 47 4 x = − 3 4 ± i 47 4

horizontal component −2 ; −2 ; vertical component −1 −1

7 + 11 i 7 + 11 i

−16 − 30 i −16 − 30 i

−4 − i 10 −4 − i 10

x = 7 − 3 i x = 7 − 3 i

x = −1 , −5 x = −1 , −5

x = 0 , 9 7 x = 0 , 9 7

x = 10 , −2 x = 10 , −2

x = − 1 ± 5 4 x = − 1 ± 5 4

x = 2 5 , − 1 3 x = 2 5 , − 1 3

x = 5 ± 2 7 x = 5 ± 2 7

x = 0 , 256 x = 0 , 256

x = 0 , ± 2 x = 0 , ± 2

x = 11 2 , −17 2 x = 11 2 , −17 2

[ − 10 3 , 2 ] [ − 10 3 , 2 ]

( − 4 3 , 1 5 ) ( − 4 3 , 1 5 )

Where the blue is below the orange line; point of intersection is x = 3.5. x = 3.5.

( 3.5 , ∞ ) ( 3.5 , ∞ )

Practice Test

y = 3 2 x + 2 y = 3 2 x + 2

( 0 , −3 ) ( 0 , −3 ) ( 4 , 0 ) ( 4 , 0 )

( − ∞ , 9 ] ( − ∞ , 9 ]

x = −15 x = −15

x ≠ −4 , 2 ; x ≠ −4 , 2 ; x = − 5 2 , 1 x = − 5 2 , 1

x = 3 ± 3 2 x = 3 ± 3 2

( −4 , 1 ) ( −4 , 1 )

y = −5 9 x − 2 9 y = −5 9 x − 2 9

y = 5 2 x − 4 y = 5 2 x − 4

5 13 − 14 13 i 5 13 − 14 13 i

x = 2 , − 4 3 x = 2 , − 4 3

x = 1 2 ± 2 2 x = 1 2 ± 2 2

x = 1 2 , 2 , −2 x = 1 2 , 2 , −2

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• Book title: College Algebra
• Publication date: Feb 13, 2015
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