One Variable Inequalities

The resources on this page will help you learn everything from the basics of inequalities, like reading and understanding them, to absolute value inequalities.    Work your way through each part, read the text, watch the videos, try the practice problems.  Give yourself time to learn this information, to really understand, and it will work out great!  If you have questions, please don’t hesitate to send an email.  

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In part of one-variable inequalities we focus on conceptual understanding and reading, writing with algebra, and interpreting graphs.  

The words and phrases, equal, not equal, greater than, less than or equal to, all hold special significance in mathematics.  Why?  Because math compares things to help us gain understanding.

However, mathematicians are notoriously … looking for efficiencies (AKA lazy writers).  As such, we don’t use the phrase greater than or equal to when writing things in Algebra.  Instead, we use a special symbol.  Here is a list of those symbols and their meaning in English.

When asked, what number is less than 5, it is easy to come up with an answer.  The problem is, one answer isn’t what is being sought.  All possible answers are being requested.  That’s where it gets a bit tricky.  There’s no way to list all of them.  Just the integers alone are infinite.  For that matter, even the numbers less than five, but more than four, are infinite.

That’s why we graph.  A graph is a picture of all of the solutions.  A solution is a value that makes a statement true.  So while we cannot list all of the numbers less than 5, we can draw a picture.  The problem is … what about 5?

Of course 5 is NOT less than itself, but what is the largest number less than 5?  Don’t say four, because there’s 4.1, and of course 4.9.  What about 4.999?  See where we are going?  

And what about the smallest answer to the question?  What is the smallest number less than 5?  Well, there really isn’t one because numbers go on forever, they’re infinite.  

Here’s what we do.  On a number line we draw the number five.  To show that the number five itself is a boundary of sorts, but is not itself an answer, we draw a circle where it is.  That circle means, “Hey, the solutions go all of the way up to, but do not include the number circled.”  Then, we simply shade in, or draw an arrow to the direction where all of the answers exist.  Anything not shaded is NOT an answer, not a solution.

Now, in some places it is not common to use a circle or a solid dot, but instead use brackets and parenthesis.  But the meaning is the same.  A circle over a number means that number is NOT a solution, but is the boundary.  A parenthesis like ( means the same thing.  

What if there was a closed circle, or a solid dot?  What would that mean?  

The solid dot here means that the number -2 is included.  The statement that would fit this graph, in English, is, “What number is greater than or equal to negative two?”

Greater than OR equal to means that -2 works.  Because -2 is equal to itself, -2 is part of the solution set.  See how that works?

How do we write that same question, “What number is greater than or equal to negative two,” in algebra?

Let’s look at the table above and use a variable.  The letter x will mean unknown value.  It will replace “What number …”  Now for the phrase, is greater than or equal to.  That’s not so bad.  That’s just ≥.    Here’s what we have.

What number is greater than or equal to negative two.

x ≥ -2

See how that works?  A key idea is to read the statements written in algebra as though they were in English.  This gives you a sense of the solutions.  If you can tie that idea into how to sketch a picture of all of the answers and boom, you’re done!

Inequalities Part 1 Lesson

Note:  This topic is often the first time students will have dealt with a number of key concepts and ideas.  They include:

  1. Graphing (including creating, reading, and interpreting)
  2. Solution sets (multiple answers)
  3. Consequences of the Real Number Line being continuous
  4. Translating from English to Algebra to graphs and back

 

As such, this is a great opportunity to help students establish a solid conceptual foundation for those topics.  The extra time and effort spent is well worth it, and will be made up with future topics when these concepts will need only maintenance.  

Big Idea

An inequality is simply a description of a relationship between a number and other numbers.  For example, x is less than 5.  This means that x could be any value less than 5 itself.  Since there are infinitely many numbers less than 5, this is the first time that graphing an answer is useful.  It provides the reader of the graph with an easily understood, visual representation of an infinite set of solutions.

Key Knowledge

Students need to be able to order numbers and also know the symbols of inequality.  

Pro – Tip 
(for students)

Whether you are given an algebraic inequality or a graph, translate the meaning into English so that you understand the relationship before carrying out the task as instructed.

 

If you’re interested in a PowerPoint, please click the link below.

Time
(minutes)

Notes

Slide #

5

Introduce the day’s topic.  Contextualize why it is important.  At the high school level many students will have seen this before and will know the short cuts.  Encourage them to dig deeper as the deeper understanding will serve them well. 

2

5

Have students discuss the question.  Guide them through as they stretch their answers and challenge their thinking.   Try not to tell them that 5 is a boundary, but have them explain why 5 is not an answer, but almost is.

3

5

This is almost the same process, but with a twist.  Here students might get confused over what OR means, confusing it with AND.  -2 is greater than OR equal to itself, but it is equal to itself. 

4

10

What we are doing here is helping students articulate answers they can figure out on their own in English, then recording those on the board.  After we establish this, we will help them learn to use the symbols in a graph, and then we will help them read and write the expressions Algebraically. 

5

10 – 15  

Students need to come away understanding that the shaded (or parts with an arrow) of a graph are the answers.  The unshaded numbers are not answers.  Here we help them understand how to do that and what the symbols mean. 

6 – 8

5 – 10

The last example is trickiest as it is a compound inequality.  We will explore compound inequalities in greater detail in the future, as we will solving inequalities.  But for now, we just want them to read, write and graph.

9

10

Now we will introduce the algebra notation and help students translate the English and graphs into inequalities.  The third example is the trickiest one again.

10

10

Closure and practice.

11 – 14

 

Homework

15

 

Assignment 1

 

  1. Answer the following questions given the graph below.

 

  • Write a statement in English that matches the graph.



  • Write an algebraic inequality that matches the graph above.



  • List all Whole Number solutions to the graph above.



  • Is the number 6 shown as a solution on the graph above?



  • Is the number 4 shown as a solution on the graph above?


 

  1. Draw a picture of the solutions to the statement: What number is greater than 0 but less than 11?

    • What is the smallest solution to the statement?



    • What is the largest solution to the statement?

 

  1. Draw a picture that matches the statement:  x = 5.




  2. Draw a picture that matches the statement: What number is less than 11 and is greater than -4?





  3. Why is it impossible to have a number that is greater than and equal to three?





  4. What does a circle mean on a graph?





  5. What are graphs and why do we use them?




  6. What is the smallest answer to the inequality: 4 < x?




  7. Is x = 2 a solution to the inequality 3x – 5x2 < 0?

           

 

The following video is for teachers.  It discusses some best practices for teaching inequalities.

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In part two we learn how to solve and graph inequalities in one variable.

Inequalities Part 2

Let’s discuss solving and graphing inequalities in one variable.  One variable means we will end up with something like x < 4, not anything like y < 3x – 2.  A few key principles we need to have under our belt are:

  • Strict inequalities, which are less than (<), and greater than (>) get an open circle on the graph.
  • Non-strict inequalities, which are less than or equal to (≤), and greater than or equal to (≥), get a close circle on the graph.
  • The line you draw signifies where solutions are. Where no line is drawn, no solutions should exist.
  • A graph is a picture of all solutions.
  • Inequalities have infinitely many solutions.
  • A solution is a value that makes a statement true.

When solving inequalities, we use inverse operations, just like when solving equations.  There is one exception.  When you multiply or divide by a negative number, the inequality changes.  Otherwise, just use SADMEP.  Let’s take a look at an example.

Solve and graph:  5x + 7 < 12

To solve we need to get the x by itself on one side of the inequality, all of the other numbers collected and combined on the other.  Since the x is already on the left, let’s keep it there.  That means we need to subtract seven from both sides, then divide both sides by five.

5x + 7 – 7 < 12 – 7

5x < 5

x < 1

This says that any number less than one is a solution.  So, -1,000 should work in the original inequality, but 1,000 should NOT work.  Let’s graph it.

Notice that we did not have to reverse the inequality here because we subtracted.  It is only when you multiply or divide that the inequality is reversed.   We won’t spend much time getting into why this happens, but enough to see that you need to switch the sign or you’ll end up with the opposite answers, and it won’t be a solution.  Let’s take a look.

13 < 1 – 4x

Here the x is on the right, and we will keep it there.  This is slightly tricky because the one is actually adding to the – 4x.   So a typical mistake here is to add 1 to both sides.  That would give you 14 < 2 – 4x, which doesn’t help us at all!

13 – 1 < 1 – 1 – 4x 

12 < – 4x

We have to divide by sides by 4.  Let’s look at what happens if we do it right.

–3 > x

This says that x is anything smaller than – 3.  So – 5 should be a solution while 0 should not be a solution.  Let’s check both and see if it works out.

13 < 1 – 4(-5) 

13 < 1 + 20

This is true because thirteen is less than twenty.

Our solution set says that anything less than -3 is a solution, which means nothing more than negative three will be a solution.  If we plug in x = 0, and get a true statement, then we have a mistake.  So when we plug in zero, we need to get a false statement.

When solving inequalities, we are saying what the answers are, and what the answers are not.

13 < 1 – 4(0)

13 < 1

This is obviously false, which is exactly what we wanted.  

Now, let’s take a look at what happens if we do not change the inequality when we divide by a negative.  For this inequality we would have gotten -3 < x.  That means that the solutions are all larger than -3, and the non-solutions are smaller than -3.  We just checked the original inequality with x = 0, and found that we had a non-solution.  We also previous showed that when x = -5, we get a solution.  This is the opposite of -3 < x.

Summary:  Solve inequalities by using inverse operations, just like when solving equations.  The only change happens if you multiply or divide by a negative.  Then you have to reverse the direction of the inequality. 

 

 

Inequalities Part 2 Lesson

Note:  This topic is often the first time students will have dealt with a number of key concepts and ideas.  They include:

  1. Graphing (including creating, reading, and interpreting)
  2. Solution sets (multiple answers)
  3. Consequences of the Real Number Line being continuous
  4. Translating from English to Algebra to graphs and back

As such, this is a great opportunity to help students establish a solid conceptual foundation for those topics.  The extra time and effort spent is well worth it, and will be made up with future topics when these concepts will need only maintenance.  

Big Idea

When solving inequalities we are finding what the answers are, and what they are not.  We can show this with a graph.   Graphs are pictures of all solutions.

Key Knowledge

To solve inequalities we use inverse operations.  The only difference between solving an inequality and solving an equation happens when we have to multiply or divide by a negative number.  Then, the inequality is changed from less to more, or more to less.

Pro – Tip 
(for students)

Whether you are given an algebraic inequality or a graph, translate the meaning into English so that you understand the relationship before carrying out the task as instructed.

 

If you’re interested in a PowerPoint, please click the link below.

Coming Soon

In part three we learn to deal with compound inequalities.  Those would be AND inequalities as well as OR inequalities.

Inequalities Part 3

Compound Inequalities

Imagine that you want to go out to the movies with some friends.  Your mom says you can go, if you do some work around the house.  Let’s consider two things she could say.  Suppose option A is that she says, “You can go out, but you have to do the dishes AND clean your room first.”  Or, option B, she says, “You can go out, but you have to do the dishes OR clean your room first.”  Which would you prefer, and why?

You undoubtedly said option B.  With “OR” we only need to fulfill one of the two options and we’re done.  With “AND” both options must be fulfilled.  What’s that got to do with inequalities?  Well, a lot.  We have to types of compound inequalities we’re going learn here.  They are called AND and OR inequalities.  Here’s an example of each.

-1 ≤ x < 2

This is an AND inequality.  The values for x are between -1 and 2.  We read it as follows.

The unknown number (x) is greater than (or equal to), -1 AND less than 2.

On a number line, this looks like this:

All of the values that are solutions are both greater than (or equal to), -1 and also smaller than 2.  There are no solutions less than -1, and no solutions more than 2.

The other inequality is an OR inequality.  These are the opposite of the AND inequalities.  Think of boat oars sticking out of the side of a boat.  Here’s an example.

x ≤ -2   OR   x > 1

There’s not a way to write these together in a single statement.  So, we leave them separate.  They look like the following graph.

 

There are no solutions in between -1 and 2.  All of the solutions are on the “outside.”

If you know how to solve inequalities, then these are too much more difficult.  Let’s start with AND inequalities.  Here’s an example.

Solve and graph the following:  -4 < 2 – 3x < 11.

You know the old golden rule of Algebra … what you do to one side you must also do to the other?  Well, in an equation, or simple inequality, we only have two sides.  Now we have three, the left, middle and right.

What you do to one side you do to all three.  Otherwise, we just use SADMEP, and remember that if you divide or multiply by a negative, the inequality symbols change.  One important thing here, that is different than many cases.  The x must stay in the middle section.  This keeps the AND inequality valid, and we don’t want to mess up the original statement.

Let’s get to it.  Here we need to get rid of the 2 and the -3.  The two is adding, so we will subtract it from all three sides, then we’ll divide all sides by -3.  When we do that we need to rewrite it.

-4 < 2 – 3x < 11

-4 – 2 < 2 – 2 – 3x < 11 – 2

-6 < – 3x < 9

2 > x > -3

If we rewrite this from smallest to greatest, we get the following.

-3 < x < 2

This graph will have open circles on -3 and 2, and the line will connect the two.  The reason they don’t continue in each direction is because the condition of AND means both conditions must be fulfilled.

We don’t graph this equation here because you should already be proficient with that.

 

The OR inequalities are a bit more like what you’re used to, the only difference being that you treat each as two separate problems.   Here’s an example.

Solve:  2x – 1 ≤ -5    OR  8x > 16

Treat in condition of the OR inequality separately.  Be sure that if you divide or multiply by a negative that you change the direction of the inequality.  Try solving this on your own.

Did you get x ≤ -2 OR x > 2?  If so, you’re doing great.

Summary:  There are two types of compound inequalities we will deal with.  The first are called AND inequalities.  Both conditions must be met.  To solve these, apply inverse operations to all three sides, keeping the x in the middle.  The OR inequalities are solving by dealing with each condition separately, like unique problems that go on the same graph.

 

Inequalities Part 3 Lesson

 

Coming Soon

In this past part we learn to about absolute value inequalities.  These tie in all of the previous sections.  When you understand this topic, you’ll really understand all of it!

Inequalities Part 4

Absolute Value

 

There are a couple of facts that you probably already know that we need to combine for this topic.  Once the connection between those facts is made it is a lot easier to understand what is happening with absolute value inequalities like the following.

|8x – 2|< 4

Let’s start with a pair of questions.

Q: What number is farther from zero, 5 or -6?

 

Q:  What number is closer to zero, -10 or 11?

 

Perhaps the key hurdle for students in learning this topic is that they don’t really understand what absolute value is.  Absolute value is a number’s distance from zero.  You were taught that |-5|= 5, which is true.  But, the reason why it is true is because -5 is five spaces from zero.

Key Fact #1:  Absolute value is a distance from zero.

 

Now, onto the second fact.  Let’s get your focus trained on the next key fact with a question.

Q:  What are inequalities?

 

Perhaps the best way to describe an inequality is by saying they’re a description or comparison of two things using the words larger or smaller.  For example, x > 4 means that the solutions, the values of x that make the statement true, are all larger than 4. 

Key Fact #2: Inequalities are comparisons that use LARGER and SMALLER.

Let’s start off by thinking about |5|.  The absolute value of 5 is the distance between 5 and zero.  As the first two questions suggested, that distance travels in both directions.  From zero to five to the right is the same exact distance as from zero to negative five, to the left. 

Let’s pull it all together now.  Try to describe what the inequality below is really saying.

x < |5|

This says that x is smaller than the distance that 5 is from zero.  Perhaps that could also be said, x is closer to zero than 5 is to zero, right?  Let’s draw a picture of all of the solutions (graph it).  We will need open circles on the graph because this is a strict inequality.  That has nothing to do with the absolute value, just the inequality symbol.

Do you remember that this is an AND inequality?   This is because all of the solutions are less than 5 and greater than -5.  If we wrote that mathematically, we would write the following.

-5 < x < 5

So, these two are the same:  x < |5| and -5 < x < 5.

What if we had this, instead?

|x|< 5

Let’s consider what it means, in English.  It says:

The distance of a number from zero is less than five.”

And this inequality, x < |5|, says:

A number is less than the distance that five is from zero.”

They’re the same!  So here’s what we have, so far.  If you have a less than absolute value inequality, what you really have is an AND inequality.  This is because we are talking about a distance that is smaller than the distance from zero that number.

Let’s take a look at two example of these inequalities, then we’ll move onto the second case, greater than.

Solve and graph:

|4x + 5|< 9

The absolute value groups together the entire expression 4x + 5.  It is a binomial, just a single number.  If we knew what x was, we could multiply it by four and then add five to find out the exact value of this number.  But, we don’t know x, so we can’t write this more simply.  Anyway, this number is closer to zero than 9 is to zero.  That’s what the absolute value inequality says. 

The number, 4x + 5, is closer to zero than the distance of 9 to zero.

Since the distance of nine to zero could be -9 to 0 as well as 0 to 9, we set up our AND inequality.

-9 < 4x + 5 < 9

From here there’s nothing new.  Inverse operations, get the x in the middle, and remember that if you multiply or divide by a negative the inequalities change.  You should be able to solve this on your own.  If not, then you need to review your notes on solving using inverse operations.  It will not be discussed here in any detail.

To solve we first subtract five from all sides, then divide by four.  This leaves us with the following.  (Be sure to reduce on the far left.)

-3.5 < x < 1

The graph will be simple, with a circle on -3.5 and 1, with the shading or line drawn between them to signify all of the values between -3.5 and 1 are the solutions.

Let’s see one more example. 

2|x + 1|– 5 < 1

 

Here we cannot distribute over the absolute value, they’re not parenthesis.  What we need to do is isolate the absolute value.  Then, we can set up the inequality to be solved.  We’ll do that by adding 5 to both sides, then dividing by 2.

|x + 1| < 3

So here we set up our AND inequality, then solve it.  You should be able to do this, only the set-up should be new material for you.  If not, you need to go back and work on solving inequalities.  We end up with the following.

-4 < x < 2

Here’s the take away:

If you end up with | x | < #, it is an AND inequality.

 

The second situation is if you have an inequality like the following.

|4 – 9x| > 8

This says that the number, 4 – 9x, is farther from zero than a distance of 8.  That’s 8 in either direction from zero.  We will use that information to set up our inequality.  But first, here’s a picture.

 

We know that 4 – 9x resides anywhere outside of the 8 and -8 on the number line.  We will use this to find out the values of x.   This is an OR inequality, and it comes from greater than (or equal to) inequalities.

Greater than can be said, mORe, and boat oars stick out of the sides of boats.

Here’s our inequality.

4 – 9x < -8   OR   4 – 9x > 8

When you solve this just remember to treat it almost as two separate statements.  They are connected by the OR, but are treated separately Algebraically.  Also, notice you’ll end up dividing by a negative here, so the inequality will switch at that point.

You should end up with x > 4/3 OR x < -4/9. 

Summary:  One of the confusing issues with absolute value inequalities is the confusion with greater distance to the left being more.  To the left are smaller numbers.  But with absolute value we are considering the distance of numbers from zero, not the value of the number itself.

If you have a greater than absolute value inequality, when you set it up, you’ll end up with an OR inequality.  Remember, greater is more, more has OR in it.  Silly, I know.  But combined with understanding why, this can be a great memory trick.

Notes and Miscellaneous Stuffs

Unusual Cases:  Distance is always positive.  Absolute value is distance.  So it is impossible to answer the question, “What distance is less than zero?”  (In place of zero could be the any negative number.)

In math that would look like |x |< 0, or |x |≤ -5.  There are no values for x that would make these true because absolute value is distance, and distance is positive.  |-3| < 0 is not true, and will not be true for any Real Number.

What about |x | ≤ 0?  Is there a value for x that would make this true?  Read it carefully, translate it into English if you can’t figure it out.

If you have the absolute value of a number that is said to be less than zero (negative), it is an impossible statement.  There are no solutions.

 

The second case is similar.  Because distance is positive, the absolute value of any number is always greater than any negative number.  In math, |x | > -4.  Any number you could think of here would be true.  If x = -1,000,000, it is true.  The absolute value of negative one-million is greater than -4.

If you have an absolute value of a number that is said to be greater than a negative number, it is true for any value of x.  It has infinite solutions.

Reference

|x | < a

|x | > a

·         This says that the distance from x to zero is less than a

·         Distance can go in either direction. 

·         So x must be closer to zero than a or –a.

·         Set up your inequality – a < x < a

·         This says that the distance from x to zero is farther (mORe) than a

·         Distance can go in either direction.

·         So x must be farther from zero than a is, or than – a is from zero.

·         Set up your inequality:  –a > x OR a < x.

Inequalities Part 4 Lesson

 

Coming Soon