inequalities practice homework 2 24 answer key

• Parallelogram
• Quadrilateral
• Parallelepiped
• Tetrahedron
• Dodecahedron
• Fraction Calculator
• Mixed Fraction Calculator
• Greatest Common Factor Calulator
• Decimal to Fraction Calculator
• Whole Numbers
• Rational Numbers
• Place Value
• Irrational Numbers
• Natural Numbers
• Binary Operation
• Numerator and Denominator
• Order of Operations (PEMDAS)
• Scientific Notation

By Subjects

• Trigonometry

Inequalities Worksheets

The worksheets given here require students to solve inequalities in the regular way and also on the number line. Some of the questions ask you to find the inequality variables’ values.

• Compound Inequalities Worksheets
• Graphing Inequalities Worksheets
• Inequality Word Problems Worksheets
• Systems of Inequalities Worksheets
• Solving Equations and Inequalities Worksheets
• One Step Inequalities Worksheets
• Two Step Inequalities Worksheets
• Linear Inequalities Worksheets
• Multi Step Inequalities Worksheets
• Quadratic Inequalities Worksheets
• Absolute Value Inequalities Worksheets
• Writing Inequalities Worksheets
• Rational Inequalities Worksheets

Related Materials

• Privacy Policy

Join Our Newsletter

© 2024 Mathmonks.com . All rights reserved. Reproduction in whole or in part without permission is prohibited.

Math Worksheets

Just another wordpress site, site navigation.

• Constant of Proportionality Worksheets
• Coordinate Graph Worksheets–Interpreting
• Equivalent Expressions Worksheets–Perimeter
• Equivalent Expressions Worksheets–Word Problems
• Integer Division Worksheets
• Number Line Worksheets–Integers
• Number Line Worksheets–Rational Numbers
• Rational Number Expressions Worksheets
• Tape Diagram Worksheets
• Analyzing Equations Worksheets
• Function Interval Worksheets
• Repeating Decimals Worksheets
• Scientific Notation Worksheets–Multiples
• Simultaneous Linear Equation Worksheets (Part I)
• Simultaneous Linear Equation Worksheets (Part II)
• Systems of Equations (How Many Solutions?)
• Transformation Effects Worksheets
• Transformation Series Worksheets
• Evaluating Expressions Worksheets
• Factoring Polynomials Worksheets

Graphing Inequalities Worksheets (Single Variable)

Solving inequalities worksheets.

• Solving Inequalities with Absolute Value Worksheets
• Order of Operations Worksheets
• Equations & Word Problems (Combining Like Terms)
• Slope of a Line from a Graph–Points Given
• Slope Worksheets (Two Points-No Graph)
• Changing One Equation
• Changing Two Equations
• Word Problems
• Multiple Choice Worksheets
• Substitution Worksheets
• Already Graphed
• Graphing Systems of Equations (Slope-Intercept Form)
• Graphing Systems of Equations (Standard Form)
• Trigonometry Worksheets
• Auto-Generated Worksheets
• Interpreting Points on a Graph
• Adding Decimals Worksheets
• Comparing Decimals-Decimal Inequalities
• Decimal Division-Integer Divisors (1 Digit)
• Decimal Division-Integer Divisors (2 Digit)
• Decimal Division-Integer Divisors (3 Digit)
• Decimal Division-Integer Divisors (MIXED Digits)
• Decimal Division-Decimal Divisors (Tenths into Tenths)
• Decimal Division-Decimal Divisors (Tenths into Hundredths)
• Decimal Division-Decimal Divisors (Tenths into Thousandths)
• Decimal Division-Decimal Divisors (Hundredths into Hundredths)
• Decimal Division-Decimal Divisors (Hundredths into Thousandths)
• Decimal Division-Decimal Divisors (Thousandths into Thousandths)
• Decimal Division-MIXED Decimal Divisors
• MIXED Decimal & Integer Divisors (1-Digit)
• MIXED Decimal & Integer Divisors (2-Digits)
• MIXED Decimal & Integer Divisors (3-Digits)
• Adding 1 Zero (Single-Digit Integer Divisor)
• Adding 1 Zero (Two-Digit Integer Divisor)
• Adding 1 Zero (Single-Digit Decimal Divisors)
• Adding 1 Zero (Two-Digit Decimal Divisors)
• Adding 2 Zeros (Single-Digit Integer Divisors)
• Adding 2 Zeros (Two-Digit Integer Divisors)
• Adding 2 Zeros (Decimal Divisors)
• Repeating Decimals (One Digit Patterns)
• Repeating Decimals (Two Digit Patterns)
• Repeating Decimals (Three Digit Patterns)
• Decimal Division Word Problem Worksheets
• Multiplying Decimals Worksheets
• Subtracting Decimals Worksheets
• Writing Decimals as Fractions Worksheets
• Checking Equation Solutions–Distributive Property
• Checking Equation Solutions–Like Terms
• Checking Equation Solutions–Variables on Both Sides
• Checking Two-Step Equation Solutions
• Solving Equations with Like Terms
• Solving Equations with the Distributive Property Worksheets
• Solving Equations with Variables on Both Sides Worksheets
• Solving Equations with Absolute Value Worksheets
• Solving Proportions
• Equations and Word Problems (Two Step Equations)
• Equations and Word Problems (Combining Like Terms) Worksheets
• Adding Fractions Worksheets
• Comparing Fractions Worksheets
• Dividing Fractions Worksheets
• Multiplying Fractions Worksheets
• Proportions & Fractions Worksheets
• Subtracting Fractions Worksheets
• Exterior Angles Worksheets
• Interior Angles Worksheets
• Parallel Lines & Transversals Worksheets
• Areas of Circles Worksheets
• Areas of Parallelograms Worksheets
• Areas of Trapezoids Worksheets
• Areas of Triangles Worksheets
• Radius Given (Using 3.14)
• Diameter Given (Using 3.14)
• Radius or Diameter Given (Using 3.14)
• Radius Given (In Terms of Pi)
• Diameter Given (In Terms of Pi)
• Radius or Diameter Given (In Terms of Pi)
• Volume of a Rectangular Prism
• Volume of a Triangular Prism
• Absolute Value of a Number Worksheets
• Absolute Value Expressions (Simplifying) Worksheets
• Absolute Value Equations Workssheets
• Absolute Value Inequalities Worksheets
• Addition Worksheets
• Division Worksheets
• Multiplication Worksheets
• Percentage Worksheets
• Square Roots
• Subtraction Worksheets
• Mean/Median/Mode/Range Worksheets
• Mean Worksheets
• Median Worksheets
• Graphs and Mean/Median/Mode Worksheets
• Absolute Value–Simplifying Expressions Worksheets
• Absolute Value Equations Worksheets
• Absolute Value Inequality Worksheets
• Probability & Compound Events Worksheets
• Probability & Predictions Worksheets
• Theoretical Probability Worksheets
• Comparing Ratios Word Problem Worksheets
• Comparing Ratios Worksheets
• Part-to-Part Ratio Worksheets
• Part-to-Whole Ratio Worksheets
• Ratio Word Problems (w/Fractions)
• Simplified Ratios Word Problem Worksheets
• Writing Ratios Word Problem Worksheets
• Writing Ratios Word Problems (missing info)
• Writing Ratios Word Problems (w/distractors)
• Writing Ratios Worksheets
• Comparing Unit Rates Worksheets
• Unit Rate Word Problem Worksheets
• Unit Rates & Graphs Worksheets
• Unit Rates & Proportions Worksheets
• Unit Rates & Tables

Looking for Something?

Popular content.

• Simplifying Expressions Worksheets
• Absolute Value Worksheets

Inequalities Worksheets

Are you looking for free math worksheets that will help your students develop and master real-life math skills?  The algebra worksheets below will introduce your students to solving inequalities and graphing inequalities.  As they take a step-by-step approach to solving inequalities, they will also practice other essential algebra skills like using inverse operations to solve equations.

Solving Inequalities Worksheet 1 – Here is a twelve problem worksheet featuring simple one-step inequalities.  Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 1 RTF Solving Inequalities Worksheet 1 PDF Preview Solving Inequalities Worksheet 1  in Your Browser View Answers 

Solving Inequalities Worksheet 2 – Here is a twelve problem worksheet featuring simple one-step inequalities.  Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 2 RTF Solving Inequalities Worksheet 2 PDF Preview Solving Inequalities Worksheet 2 in Your Browser View Answers

Solving Inequalities Worksheet 3 – Here is a twelve problem worksheet featuring two-step inequalities.  Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 3 RTF Solving Inequalities Worksheet 3 PDF Preview Solving Inequalities Worksheet 3 in Your Browser View Answers

Solving Inequalities Worksheet 4 – Here is a twelve problem worksheet featuring one-step inequalities.  Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 4 RTF Solving Inequalities Worksheet 4 PDF Preview Solving Inequalities Worksheet 4 in Your Browser View Answers

Solving Inequalities Worksheet 5 – Here is a twelve problem worksheet featuring two-step inequalities.  Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 5 RTF Solving Inequalities Worksheet 5 PDF Preview Solving Inequalities Worksheet 5 in Your Browser View Answers

Graphing Inequalities Workheet 1 –  Here is a 15 problem worksheet where students will graph simple inequalities like  “ x < -2″  on a number line. Graphing Inequalities Worksheet 1 RTF Graphing Inequalities 1 PDF View Answers

Graphing Inequalities Workheet  2 –  Here is a 15 problem worksheet where students will graph simple inequalities like  “ x < -2″  and “ -x > 2″  on a number line.  Be careful, you may have to reverse one or two of the inequality symbols to get the correct solution set. Graphing Inequalities 2 RTF Graphing Inequalities 2 PDF View Answers

Graphing Inequalities Workheet  3 –  Here is a 12 problem worksheet where students will both  solve  inequalities and  graph  inequalities on a number line.  This set features one-step addition inequalities such as   “x + 5 > 7”. Graphing Inequalities 3 RTF Graphing Inequalities 3 PDF View Answers

Graphing Inequalities Workheet  4 –  Here is a 12 problem worksheet where students will both  solve  inequalities and  graph  inequalities on a number line.  This set features one-step addition and subtraction inequalities such as   “5 + x > 7”  and  “x – 3″ < 21”. Graphing Inequalities 4 RTF Graphing Inequalities 4 PDF View Answers

Graphing Inequalities Workheet  5 –   Here is a 12 problem worksheet where students will both  solve  inequalities and  graph  inequalities on a number line.  This set features two-step addition inequalities such as   “2x + 5 > 15” . Graphing Inequalities 5 RTF Graphing Inequalities 5 PDF View Answers

Graphing Inequalities Workheet  6 –  Here is a 12 problem worksheet where students will both  solve  inequalities and  graph  inequalities on a number line.  This set features two-step addition and subtraction inequalities such as   “2x + 5 > 15”  and “ 4x -2 = 14. Graphing Inequalities 6 RTF Graphing Inequalities 6 PDF View Answers

Absolute Value Inequality Worksheets (Single Variable)

Absolute Value Inequality Worksheet 1 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are one-step inequalities with mostly positive integers. Absolute Value Equations Worksheet 1 RTF Absolute Value Equations Worksheet 1 PDF View Answers

Absolute Value Inequality Worksheet 2 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.   These are one-step inequalities where you’ll need to use all of your inverse operations knowledge. Absolute Value Equations Worksheet 2 RTF Absolute Value Equations Worksheet 2 PDF View Answers

Absolute Value Inequality Worksheet 3 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are two-step inequalities where you’ll need to use all of your inverse operations knowledge. Absolute Value Equations Worksheet 3 RTF Absolute Value Equations Worksheet 3 PDF View Answers

Absolute Value Inequality Worksheet 4 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are two-step inequalities that can get quite complicated.  A nice challenge for your higher-level learners. Absolute Value Equations Worksheet 4 RTF Absolute Value Equations Worksheet 4 PDF View Answers

20 Comments

This web is good

Thanks for your support.

i like it. Its very helpful.

I appreciate the feedback. Good luck with your math endeavors!

I like this website. It is very helpful for students.

Thanks. I’ll be adding more worksheets soon.

Interesting questions. Very good.

Thanks. I’m glad you enjoyed them.

This is a great site! Thanks so much for helping me teach this concept to my 5th graders!

Mr. Colwell

I’m glad that I could help. Thanks for visiting the site and come back often!

Kaity Fanara

Very helpful, thanks!

You’re very welcome. I plan to add some Graphing Inequalities Worksheets soon. Thanks for your support.

c’moon! so easy give me something like, ( m-2 ) x2 – (m+5) x + m-8 = 0, if it has 2 equal roots, what’s m? i need this kind, where can i find

That’s a pretty good one. Let me see what I can do.

Mr. Lindugani Mlilile

Thanks for your website which helps me teach my students of form one in Tanzania. I always teach my students on how solve the topic of inequality. Please add more topic so that my students will continue to enjoys!

It’s nice to hear from our international visitors. I’m always adding new content. Check back often!

It is helpful but I can’t see the answer sheet for all of them.

wish their was some fractions inequalities and equalities

however work sheets have helped me tremendously just the one little flaw of not having fraction problems.

I LOVE THIS WEBSITE. I USE IT EVERY DAY TO STUDY FOR TEST OR QUIZ FOR SCHOOL.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

• Skip to main content
• Skip to primary sidebar

CLICK HERE TO LEARN ABOUT MTM ALL ACCESS MEMBERSHIP FOR GRADES 6-ALGEBRA 1

Maneuvering the Middle

Student-Centered Math Lessons

Teaching One- and Two-Step Inequalities

What I love so much about inequalities are the infinite solutions. For so much of students’ previous math experience, there is one exact answer. When it comes to inequalities, it is fun to push students to think of their answers beyond “x<4” and brainstorm all of the infinite solutions that x can be and then connect it to the graph.

• Can x be 3.9999?
• Can x be 4?
• Can x be -100? 

Below, I have outlined a few ideas and things to consider when planning to teach one- and two-step inequalities.  

Let’s take a look at the standards, shall we?

Many of my recommendations for solving one or two-step inequalities are the same as solving one and two-step equations. This post talks about how using algebra tiles is my favorite practice. 

Something I did not learn until I was teaching is the why behind changing the sign when you divide or multiply by a negative number. As

You can do whatever you want to an inequality as long as it is done to both sides, it will remain a true statement.

This is true of inequalities with one exception. If you multiply or divide by a negative number, the inequality becomes untrue.

While this mistake still persisted, I found that my students did much better than previous groups in remembering to flip the sign.

Able to Read an Inequality Statement

Another big misconception that I have found to be true is the “alligator.”  Sure, “the alligator eats the bigger number” works when you are comparing 3 and 7, but what about when it’s 3 and -7 or 4x and 6?   Students need to be able to read an inequality statement and explain what it means in terms of the numbers around it.  They need to feel confident choosing a number for x to test the inequality.  

I have found the students struggle identifying “<” as less than and “>” as greater than, so when they do solve an inequality and they are left with x<4, they don’t actually know what that symbol means. 1<4 is something that a student can say but replace one of the numbers with a variable and the inequality becomes unclear. This is definitely a trick, but showing students how the less than symbol can be slightly rotated to be an L will allow them to actually state, “x is less than 4.”

Using Number Lines

The graph can seem like one more thing to do, but the solution(s) and the thinking lies within the graph.  The number line representation is a perfect visual to actually get students talking about the potential solutions.   Some questions to get students thinking:

• What is a value of x that is in the solution set?  Is there another?
• Why is _____ not a solution to inequality?  What other numbers would not be a solution?
• Describe the process for graphing the solution.
• Given the graph, what inequality statement best describes it?

When we are given a graph and are trying to figure out what inequality is the match, I avoid the trick that the arrows should match the inequality sign. Instead I might ask:

• What are the potential solutions? 
• Are the potential solutions greater than or less than the given amount?
• When I substitute a solution into X, is the original inequality true?

Common Misconceptions

• Forgetting to change the inequality sign when dividing by a negative number
• General confusion when the answer is “4 > x” or written with the constant on the left
• Unclear of what number is considered a solution
• Unable to make connections between the inequality and the actual solution on the number line
• Writing inequality statements from situations when the terms “less than” or “greater than” are not included

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference.  They can easily be created before the lesson or as you are teaching, depending on the content.   This example includes the emphasis on vocabulary, as students tend to struggle with writing inequalities.

The real-life example that I think students have the most understanding of is related to movie ratings and/or height requirements for roller coasters. These very real-life experiences will help students understand the terms minimum means greater than or equal to. The minimum height you can be to ride this roller coaster is 48 inches. That means you can be 48, 49, 50 … inches to ride. Similarly, a person must be the minimum age of 17 to purchase an R-rated movie ticket.

Maximum was also another term that confused my students. Again, relating it to something that students experience daily seemed to do the trick. I would use seat belts as an example. I would ask students how many seat belts were typically in a car (or their car). Then I would say that the maximum number of people that could safely ride in said car was, let’s go with 5, which means 5 people or fewer could ride in the car. Maximum meant less than or equal to.

IDEAS FOR STRUGGLING STUDENTS

• Bring out the manipulatives – algebra tiles, pattern blocks, etc – have students group like items
• Break down the steps into a simple checklist
• Go back to positive whole numbers to see if students are struggling with the concept or the mathematical skills
• Give students a possible solution and ask them to work backwards

Hopefully, this gives you some ideas for teaching one- and two-step inequalities or even insight as to what knowledge your students are coming with.  I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Be sure to check out these resources for inequalities:

• 6th Equations and Inequalities Unit (TEKS)
• 6th Equations and Inequalities Unit (CCSS)
• 6th Grade Equations and Inequalities Activity Bundle
• 7th Grade Equations and Inequalities Unit (TEKS)
• 7th Grade Inequalities Unit (CCSS)
• 7th Grade Inequalities Activity Bundle

How do you teach inequalities?

Getting Started with Algebra Tiles

Check out these related products from my shop.

Reader Interactions

August 25, 2022 at 4:41 am

That’s a wonderful technique of teaching children about one and two step inequality topics. Kids require to understand math concepts properly before dealing with problem solving. Thank you very much. Keep sharing this more and more! Devoir math

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

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra
• Publication date: Feb 13, 2015
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra/pages/chapter-2

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.