### Solving Linear Inequalities in One Variable

- A solution to an inequality is usually a range of values that will make the statement true.
- We can describe a range of values using inequalities, interval notation, a number line, or set builder notation.
- To solve a linear inequality you may add and multiply the same amount to both sides of the inequality. If you multiply or divide by a negative number we flip the inequality symbol.

### Solving Linear Inequalities in One Variable

- x − 14 + 14 ≤ 22 + 14

- - 6( − [k/6] ) ≤ − 4( − 6)

- [( − 30)/( − 6)] < [( − 6n)/( − 6)]

- r > - 4( - 12)

- a ≥ − 3( − 13)

- 3x ≥ − 36

- 5x < 135

- − 4y > 64

- − 2u ≤ 42

- 6r > 24

- − 8p < 64

- − 3c + 15 − 6 ≥ 2c + 8
- − 3c + 9 ≥ 2c + 8
- − 3c ≥ 2c − 1
- − 5c ≥ − 1

- − 4h − 8 − 6 < 8h + 10
- − 4h − 14 < 8h + 10
- − 4h < 8h + 24
- − 12h < 24

- 6f − 10 + 4 ≥ 3f + 12 − 10
- 6f − 6 ≥ 3f + 2
- 6f ≥ 3f + 8
- 3f ≥ 8
- f[8/3]

- 4i − 16 + 1 < 6i + 12 − 2i
- 4i − 15 < 4i + 12

Not True

Empty Set

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Solving Linear Inequalities in One Variable

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Objectives
- Solving Linear Inequalities in One Variable
- Inequality Expressions
- Linear Inequality Solution Notations
- Inequalities
- Interval Notation
- Number Lines
- Set Builder Notation
- Use Same Techniques as Solving Equations
- 'Flip' the Sign when Multiplying or Dividing by a Negative Number
- 'Flip' Example
- Example 1
- Example 2
- Example 3

- Intro 0:00
- Objectives 0:08
- Solving Linear Inequalities in One Variable 0:37
- Inequality Expressions
- Linear Inequality Solution Notations
- Inequalities
- Interval Notation
- Number Lines
- Set Builder Notation
- Use Same Techniques as Solving Equations
- 'Flip' the Sign when Multiplying or Dividing by a Negative Number
- 'Flip' Example
- Example 1 8:54
- Example 2 11:40
- Example 3 14:01

### Algebra 1 Online Course

### Transcription: Solving Linear Inequalities in One Variable

*Welcome back to www.educator.com.*0000

*In this lesson we are going to take a look at solving linear inequalities in one variable.*0003

*To solve linear qualities, you will see that the process is not so bad. *0010

*But we are willing to take a little bit of time just to highlight the difference between inequalities and equations, *0013

*after looking at the difference between the two and that will cause a little bit of a problem.*0020

*You have to be careful on how we describe our solutions for inequalities.*0025

*Then we will get into a bunch of different examples on how we can finally solve these linear inequalities. *0030

*Let us get started.*0035

*When you are looking at an inequality versus an equation and you try to figure out what is the difference.*0040

*One of the main differences, of course you will see a different symbol in there, one of our inequality symbols.*0045

*It is these guys that we saw earlier.*0051

*We have our greater than symbol, greater than or equal to symbol, our less than symbol or less than or equal to symbol.*0054

*It is a little bit more than just having an additional symbol in there that makes this different. *0063

*What makes an inequality different from an equation has to do with the solutions that you get out of them.*0068

*To highlight this I will go over some quick examples of equations and get to one of those inequalities.*0075

*Consider this first one here, 3x = 15.*0081

*If you are going to solve that, it would not take probably that long.*0087

*Just simply divide both sides by 3 and you get that x = 5. *0091

*Now what that says the only number, the only one out of all possible numbers that will make that equation true is just 5.*0099

*It is like this little isolated point out on the number line.*0106

*Some equations might actually have more than one solution, for example, this next one x ^{2} = 9. *0110

*If you are going to look for numbers that will make that equation true, you will actually find two of them.*0117

*When x = 3 and you square it you will get 9, but also when x = -3 and you square it you still get 9.*0123

*You know it is a little different that you have two solutions, but both of the solutions those are the only two things that will make it work.*0132

*It is like two little isolated points on your number line.*0138

*To contrast that to this first inequality over here, 4x is greater than or equal to 8.*0142

*If you try to figure out what numbers make that true, you would end up with a huge, huge list of numbers that work.*0149

*Let us start picking out some examples. *0157

*Also, what if I said that x was equal to 2.*0159

*If you substitute that in, you would have 4 Ã— 2 is greater than or equal to 8, *0163

*which would simplify to 8 is greater than or equal to 8, which is, of course true.*0167

*But it is not the only thing that works out. *0174

*You could also put in 3, 4, you can get a little creative and put in some other ones like 4 1/2, and all of these would be solutions.*0176

*Trying to list out solutions for an inequality is simply not the way to go.*0191

*The reason for that is if you look at the numbers that would make an inequality true, you will get an entire range of values.*0196

*So, not just in a little isolated point, they are in whole ranges of values on our number line.*0204

*Since we are after a whole range of values other than these isolated points, we are going to have to be a little bit careful on how we describe those.*0211

*In fact, there are many ways that you can describe the solution to an inequality.*0223

*Some you seen before and some might be a little new, let us go over them.*0227

*The first way that you can describe the solution is to simply use our inequality symbols that will be like this guy down here.*0231

*Maybe Iâ€™m trying to describe all the numbers that are less than 4, I can simply write x is less than 4.*0239

*We can also use our interval notation.*0246

*This is a good way to mimic the number line and that it gives us the lowest number or the starting point and the endpoint. *0250

*It also gives us information on whether we should include the number or not.*0262

*Since this one has parentheses, I know that the 4 is not included.*0266

*It is simply a different way of writing the same information.*0273

*This interval says I'm looking at all the values that are less than 4. *0276

*We could also describe our solutions a little bit more visually using a number line.*0283

*The way we do this is we have a number line and we shade in the solution.*0288

*To indicate whether the endpoint is included or not, you can use an open circle or close circle.*0300

*The open circle here means it is not included, so I would have used a close circle if I had x is less than or equal to 4.*0310

*And one last way that you might see, I do see this in a few Math books is known as set builder notation. *0323

*It looks a little clunky and that we have a couple of curly brackets thrown in there *0329

*but let me show you how you can interpret the set builder notation.*0335

*First of all, we identify some sort of variable at the very beginning. *0341

*We are looking for all x values.*0346

*And after that you will notice that we borrow our inequality. *0352

*This is the rule of all of the numbers that we will include.*0356

*That little line you see in between, we say that is L and it is just the divider or it separates both those sections. *0370

*The way you would read the set builder notation is the set of all xâ€™s such that x is less than 4.*0384

*We know that we are building for x values and we know which x values would get in there.*0393

*Iâ€™m not going to focus a whole lot on that set builder notation but you will definitely see me use some inequalities, some number lines *0397

*and some interval notation just to describe our solutions.*0404

*Now that we know little bit about what makes an inequality different, we know little bit more on how to describe the solutions, *0410

*let us get into how we can actually start solving these things.*0416

*The good news is when it comes to solving linear inequality, we use the same techniques as we do with solving a normal linear equation.*0421

*There is one additional rule that you want to be aware of.*0432

*When you multiply or divide by a negative number then you want to flip the inequality sign.*0435

*Now that rule applies to multiplying and dividing, so be careful not to try and use it if you are only adding and subtracting.*0444

*Use it only for multiplying and dividing.*0453

*I often get lots of questions like why is it that you have to flip the sign when your doing the multiplication and division by negative numbers.*0458

*I think a quick example will help you understand why that is.*0465

*Let us take a nice inequality like 2 < 3, so seems pretty simple. *0469

*We know that 2 < 3 but watch what happens if I multiply both sides by a negative number.*0476

*-2 and -3, now which one is less than the other one?*0483

*If we take a peek at our number line, you will see that it is the -3 that is now the smaller number since it is on the left side of -2.*0491

*This means that if we want to preserve the trueness of our statement, we also must flip that sign to compensate for that negative that we threw in there.*0510

*It is easy to see with numbers like this but in a moment when we start dealing with xâ€™s and unknowns, *0520

*we still have to remember to flip the sign when we divide or multiply by that negative number.*0525

*I think we have all the information we need.*0534

*Let us go ahead and get into the solving process and see if we can tackle some of these inequalities.*0537

*We want to solve the following inequality then write our answer using a number line and using interval notation.*0543

*Okay, when I solve an equation usually I try and isolate the x's and I'm going to do the same thing here. *0549

*I will do that by first moving the 5 to the other side, 3x < 11 - 2x and then we will go ahead and we will add 2x to both sides. *0557

*5x < 11, let us go ahead and divide both sides by 5.*0578

*I will get that x < 11/5.*0593

*This could represent my solution using just the inequality symbol. *0597

*Now I want to go and have a look at the number line and its interval notation.*0601

*Here is a nice quick sketch of a number line. *0615

*I'm looking to shade in all values that are less than 11/5.*0620

*If I want to rewrite that, 5 goes into 11 twice with the remainder of one, so it is the same as 2 1/5.*0625

*It is a little bit greater than 2, but not much.*0638

*Notice how Iâ€™m using an open circle there because our inequality is strict. *0643

*We do not want to include that value.*0650

*I will shade in everything that is less than my 2 1/5.*0653

*This number line represents all of the solutions to my inequality, I could use anything on that side and it worked. *0662

*As long as we have our number line, let us go ahead and also represent our solution using interval notation.*0670

*We want to think of where we start on the left side, it looks like we are starting way down at negative infinity.*0680

*We go all the way up to 11/5 but I will use a parenthesis since we do not include that value.*0686

*I have represented my solution now in 3 different ways. Not bad.*0694

*Let us look at another, for this one we are looking to solve 13 - 7x = â€“ 4.*0697

*Let us start off by trying to get those x's isolated and we will do that by first getting them together. *0710

*I'm going to move this 10x to the other side, so I'm subtracting here. *0716

*I do not have to worry about flipping any signs nothing like that yet, 13 - 17x = -4.*0722

*Let us go ahead and subtract 13 from both sides, we have -17x = -17.*0735

*Be very careful in this next step.*0750

*I need to get the x all by itself, in order to do that I will divide by -17.*0751

*I will do all of the algebra normally -17 Ã· -17 = 1. *0763

*I'm also going to remember what to do with that sign, we are going to flip it around.*0770

*I have it x = 1.*0777

*Let us get to our number line and go ahead and represent the solution.*0784

*I have the number 1,we will make a nice solid circle since this is or equal to *0793

*and we will go ahead and shade in everything less than that since it says x < 1.*0804

*Now that I have my number line, let us represent this using our interval notation.*0811

*On the left side we are starting at way down to negative infinity and going all the way up to 1.*0815

*We want to include the one so we will use a bracket.*0821

*Let us try one more example.*0838

*In this last one I picked a little bit more of a word problem so you can see that inequalities are important even in applications.*0843

*This one says that Joanna is hiring a painting company in order to get her house painted*0850

*and the company has two plans to choose from, you can choose plan A.*0855

*In plan A they charge you a flat fee of $250 and $10 for every hour that it takes to paint the house.*0861

*For plan B, they do not charge a flat fee but they will charge $20 per hour.*0869

*The question is when will plan B be more expensive than plan A?*0874

*Pause for a moment and think why would we be using an inequality here?*0878

*Why do not we just set up an equation?*0882

*What I'm looking for is not one specific time, but all times when plan B will be more expensive.*0885

*It makes a little bit more sense to go ahead and set up an inequality for this word problem.*0891

*Let us go ahead and hunt down both of our situation.*0896

*Let us have plan A and let us go ahead and have plan B.*0899

*We need an unknown here, let us say x is the time to paint the house.*0907

*Plan A cost a flat fee of $250 + $10/hour.*0924

*$10/ hour is like our variable cost, that would be 10x, flat fee does not vary so +250.*0929

*That expression will just keep track of the cost for plan A depending on how many hours it takes.*0938

*Plan B has only a variable cost of $20/hour, I will just say 20x.*0944

*Now comes the big question, when is plan B more expensive than plan A.*0951

*B would be more expensive, I would use an inequality symbol of plan A < plan B *0958

*and that is exactly how we will connect our expressions as well. *0964

*10x + 250 < 20x and to completely answer this problem we just have to solve the inequality.*0967

*Let us work on getting our x's together by subtracting 10x from both sides.*0976

*250 < 10x, now we will divide both sides by 10.*0986

*25 < x, now it is time to interpret exactly what we have here.*0999

*x represents the time that it will take to paint the house.*1006

*What I can see here is that anytime my time is more than 25 hours then I can be sure that plan B will be more expensive, there you have it. *1011

*Also, be careful when setting up and solving these inequalities.*1021

*Remember to go ahead and flip your inequality anytime you multiply or divide by negative number.*1024

*Thank you for watching www.educator.com.*1031

1 answer

Last reply by: Professor Eric Smith

Thu Jul 17, 2014 10:44 AM

Post by Francis Tremblay on July 15, 2014

IS THERE ANYWAY TO DOWNLOAD VIDEOS I'M USING A CHROMEBOOK?

1 answer

Last reply by: Professor Eric Smith

Mon Jun 16, 2014 3:22 PM

Post by patrick guerin on June 14, 2014

Thank you for all of your information on algebra 1, it has really helped me keep up on my mathematics over the summer. Thanks again!