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### Compound Inequalities

- When using the connector AND, both conditions must be satisfied. (Intersection)
- When using the connector OR, at least one of the conditions must be satisfied. (Union)
- A compound inequality is more than one inequality connected using AND or OR.
- Some inequalities can be put together into three parts. For these, whatever you do to one piece, you must do to the other two.

### Compound Inequalities

3x − 2 < 74x + 6 > 10

- 3x − 2 < 7
- 3x < 9
- x < 3
- 4x + 6 > 10
- 4x > 4
- x > 1

- 7s − 5 ≤ 16
- 7s ≤ 21
- s ≤ 3
- 3s + 10 ≥ 22
- 3s ≥ 12
- s ≥ 4

- 6d + 2 > 8
- 6d > 6
- d > 1
- 7d − 5 < 9
- 7d < 14
- d < 2

7k − 7 > 35

- 3k − 4 ≤ 8
- 3k ≤ 12
- k ≤ 4
- 7k − 7 > 14
- 7k > 21
- k > 3

4n − 1 ≥ 19

- 2n − 3 < 5
- 2n < 8
- n < 4
- 4n − 1 ≥ 19
- 4n ≥ 20
- n ≥ 5

4 < n ≥ 5

5m − 4 ≥ 51

- 8m − 2 ≤ 22
- 8m ≤ 24
- m ≤ 3
- 5m − 4 ≥ 51
- 5m ≥ 55
- m ≥ 11

3 ≥ m ≥ 11

- − 9 < 6x − 7 and 6x − 7 < 11
- − 9 < 6x − 7
- − 2 < 6x
- − [2/6] < x
- − [1/3] < x
- 6x − 7 < 11
- 6x < 18
- x < 3

- − 34 ≥ 6x + 26x + 2 > 26
- − 34 ≥ 6x + 2
- − 36 ≥ 6x
- − 6 ≥ x
- 6x + 2 > 26
- 6x > 24
- x > 4

− 2h − 11 > 9 or − 5h + 7 ≤ − 13

- − 2h − 11 > 9
- − 2h > 20
- h <− 10
- − 5h + 7 ≤ − 13
- − 5h ≤ − 20
- h ≥ 4

- − 5x − 4 ≥ 11
- − 5x ≥ 15
- x ≤ − 3
- − 3x + 12 < 36
- − 3x < 24
- x >− 8

*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

### Compound Inequalities

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 0:00
- Objectives 0:07
- Compound Inequalities 0:37
- 'And' vs. 'Or'
- 'And'
- 'Or'
- 'And' Symbol, or Intersection
- 'Or' Symbol, or Union
- Inequalities
- Example 1 6:22
- Example 2 9:30
- Example 3 11:27
- Example 4 13:49

### Algebra 1 Online Course

### Transcription: Compound Inequalities

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

*In this lesson, we are going to take care of compound inequalities.*0002

*In compound inequalities we will look at connecting inequalities using some very special words and, and or.*0010

*We will have to first recognize some subtleties between using both of these words.*0017

*Blabbering about some nice new vocabulary, we will learn about union and intersection.*0023

*Once we know more about how to connect them then we can finally get to our compound inequalities and how we can solve these.*0029

*In some situations, you have more than one condition and they could be connected using the word and or they could be connected using or.*0040

*To see the subtle difference between using one of these things, let us just try them out on a list of numbers.*0050

*I have the numbers 1 all the way up to 10 and we will see what number should be included in the various different situations below.*0056

*Let us see if we can first take care of all the numbers greater than 3 and less than 7.*0066

*Notice in this one I’m using that word, and.*0071

*Numbers that are greater than 3, we will have 4 or 5 and 6 and all of these are also less than 7.*0075

*The things I will include in my list 4, 5, and 6.*0083

*Let us think on how we will approach this problem a little bit different for the next one.*0090

*Maybe list out all numbers that are greater than 4 or they are less than 2.*0095

*The numbers greater than 4, that would we 5, 6, 7, 8, 9, and 10 and the numbers that are less than 2, 1.*0101

*Notice how in that situation I was looking for 2 things.*0112

*I was looking for all numbers that were greater than 4 and I have highlighted those first.*0115

*And then I went back and I looked for ones that were less than 2.*0119

*Let us give it a few little marks here and try the third situation.*0127

*All numbers greater than 6 and they are less than 3.*0132

*That is a tough one to do because I can go ahead and highlight the numbers greater than 6, that will be 7, 8, 9, 10 but they are not less than 3.*0137

*I can not include them.*0148

*Notice how in that situation since I'm dealing with and, it is like I have to satisfy both of these conditions must be greater than 6 must be less than 3.*0150

*If it does not satisfy both of them then I cannot include them.*0164

*Let us try another one and see how it works out.*0167

*All numbers greater than 4 so I have 5, 6, 7, 8, 9, 10 or they are less than 7.*0170

*What numbers less than 7 would be 6, 5, 4, 3, 2,1.*0178

*All the numbers fall into one category or they have fallen to the other one.*0183

*In fact, using or seems like it is a little bit more relaxed as long as it satisfies one of my conditions, then I will go ahead and include it in my list.*0188

*In fact, that highlights the difference between using and using the word or to connect these 2 conditions.*0197

*Let us make it even more clear.*0204

*When you use and to connect two conditions, then you must have both conditions met in order to include it.*0207

*However, when you are using or, then as long as you have one of the conditions met then you can go ahead and include that.*0214

*The way you will see and and or used is when we combine our intervals in our solutions.*0222

*If an object meets both conditions using our and connection, then it is said to be in the intersection of the conditions*0230

*and we can use the symbol to connect those.*0239

*Some people might think of this as and but it stands for the intersection.*0242

*On the flip side, if an object meets at least one of the conditions using the word or,*0253

*then it is said to be in the union of the conditions.*0257

*We will use the symbols, think of or and union.*0261

*You will see these symbols for sure and watch how we connect our solutions using either and or or.*0267

*We know a little bit more about the connections, let us get into how we can actually solve these things.*0277

*We will connect these things using and or or and the way you go about solving is you can actually solved each of the inequalities separately.*0285

*Just take care of one at a time and the part where these and or or come into play is when we want to connect together our solutions.*0294

*There are some situations where we can actually connect the inequalities at the very beginning.*0305

*One of those situations is when you are dealing with and and 2 of the parts are exactly the same.*0310

*Just like this example that I have highlighted below.*0316

*I have 5x < or = 3 + 11x and I also have a -3 + 11x on the other side over here.*0321

*We are connected using and.*0329

*I'm going to put these together into what is known as a compound inequality.*0332

*You will notice that everything on this side of the inequality comes from the left side.*0336

*Everything on the other side of inequality over here comes from this inequality.*0344

*It encapsulates both of them at the same time.*0351

*If you do connect one like this, the way you end up solving it is just remember that whatever you do to one part, do it to all 3 parts.*0354

*If you subtract 3, subtract 3 from all parts.*0362

*This also includes if you have to flip a sign.*0366

*If you multiply by a negative number then flip both of those inequality signs that are present then you should be okay.*0368

*Just remember if you want to solve each one separately, that works to.*0377

*Let us see some of our examples, see the solving process in action.*0383

*We want to solve the following inequality and of course write our answer using a number line and using interval notation.*0388

*I can see I have 2x -5 < or = - 7 or 2x – 5 > 1.*0395

*I’m going to solve these just separately, just take care of one at a time.*0403

*Looking at the left, I will add 5 to both sides, 2x > or = -2.*0406

*Now I can divide both sides by 2 and get that x < or = -1.*0418

*There is one of my solutions, let us focus on the other one.*0429

*You will add 5 to both sides of this inequality, 2x > 6, now divide by 2 and get that x > 3.*0434

*I have 2 intervals and I will be connecting things, let us drop this down using or.*0450

*I want to think of all the numbers that satisfy one of these, or satisfy the other one.*0458

*As long as they satisfy one of these conditions, I will go ahead and include it in my overall solution.*0463

*Let us take this a bit at a time.*0469

*I will do a little number line here, a number line here and I work on combining them into one number line.*0472

*First I can look at all the numbers that are less than or equal to 1.*0481

*Here is -1, I'm using a solid circle because it is or equals to and we are less than that so we will shade in that direction.*0489

*For the other inequality, I'm looking at numbers like 3 but not included or greater so I’m using an open circle there.*0501

*What will I put on my final number line is all places where I shaded at least once.*0509

*Let us see how that looks, I have my -1 and everything less and I have 3 or everything greater.*0523

*This number line which has both of them shows one condition or the other, and I can include both.*0535

*Let us go ahead and represent this using our interval notation.*0543

*We have everything less than - 1 and we include that -1 and we have everything from 3 up to infinity.*0546

*Since we are dealing with or, let us go ahead and use our union symbol to connect the two.*0558

*I have the many different ways that you can represent the solution for this inequality.*0563

*Let us take a look at another one, solve the following inequality and write your answer using a number line and interval notation.*0571

*This is a special inequality, this is one of our compound inequalities because it have 1, 2, 3 different parts to it.*0578

*As long as you remember that whatever we do to one part, we should do it to all three,I think it will turn out okay.*0585

*Let us work on getting that x all by itself in its particular part.*0591

*We will go ahead and subtract 5 from all three parts and get – 8 < 2x is < or = 2.*0597

*Let us divide everything by 2.*0612

*-8 ÷ 2 = -4, x < or = 1.*0620

*In this one I am looking for all values between -4 and 1.*0628

*I think we can make a number line for that.*0633

*Okay, I need to shade in everything in between -4 and all the way up to 1.*0649

*It looks like the -4 is not included, I will use a nice open circle, but the 1 is included, so we will shade that in.*0657

*Now that we have our number line, we can describe this using our interval notation.*0667

*We are starting way down at -4 not included and going all the way up to , that is included.*0671

*In some of these inequalities, you have to be careful on which conditions it satisfies.*0689

*If you end up with no numbers that simply do not work, watch how that turns out.*0695

*We want to solve the following inequality and write our answer using a number line and interval notation.*0701

*I have 4x + 1 > or = -7 or -2x + 3 > or = 5.*0706

*Let us begin by solving each of these separately.*0716

*With 1 on the left, I will start by adding 1 to both sides, 4x < or = -6 and now divide both sides by 4.*0721

*We are reducing that, I get that x > or = -3/2.*0737

*We will set that off to this side and solve the other one.*0744

*Let us subtract 3 from both sides and we will go ahead and divide both sides by -2.*0749

*Since we are dividing by negative, I’m going to flip my inequalities symbol.*0758

*This one is very interesting.*0766

*Notice how I'm looking for numbers that are less than or equal to 1, but I’m also looking for numbers that are greater than or equal to -3/2.*0768

*If I look at where those two are located, here is -3/2 and here is 1.*0779

*You will see that we get actually all numbers because I'm looking for things that are greater than or equal to -3/2,*0785

*that will shade everything on the right side of that - 3/2.*0792

*Everything less than or equal to 1 would shade everything in the other direction.*0796

*It will end up shading the entire number line.*0801

*We will say we would not include all numbers from negative infinity up to infinity.*0805

*I might make a note, all real numbers.*0813

*Watch for a very similar situation to happen when we deal with and.*0824

*Solve the following inequality and write our answer using a number line and interval notation.*0830

*I have two inequalities and a bunch of different x's, let us work on getting them together first.*0837

*I'm going to subtract an x from both sides on the left side here, 1x < -5.*0842

*I can add up 8 on both sides and we will get that x < 3.*0851

*Let us go over to the right and see what we can do there.*0861

*I will subtract 15 from both sides, that will give me –x < -10.*0864

*Now divide both sides by -1, since we are dividing by a negative number, let us flip out the inequality sign.*0871

*I have x < 3 and x > 10.*0882

*This one actually proves that when we are dealing with and, it must satisfy both of these conditions in order to be included.*0889

*If you start thinking what numbers are less than 3 and also greater than 10, you will that you do not get any number.*0896

*You can not be both things at the same time.*0902

*They are both in completely different spots on a number line.*0906

*Let us make another one.*0913

* We can not shade in numbers that are both less than 3 and greater than 10.*0917

*What is that mean to our solution then?*0942

*I mean what can we put down?*0943

*This is where we say there is no solution.*0947

*Be careful on using those connectors and how it affects your solution.*0953

*Remember that when using the word or, it must satisfy one of the conditions.*0957

*If it does, go ahead and include it in your solution.*0961

*However, when using the word and, it must satisfy both of those conditions before you can include it in your solution.*0964

*Thanks for watching www.educator.com.*0971

0 answers

Post by Karlin Hilai on February 17, 2017

4x+1 is greater than or equal to -7 should be x is equal to or greater than -2.

1 answer

Last reply by: Professor Eric Smith

Thu May 29, 2014 10:47 AM

Post by Mukesh Jain on May 24, 2014

There is a "subtle" typo, as subtle was spelled subtitle :)

2 answers

Last reply by: sherman boey

Wed Aug 13, 2014 9:26 AM

Post by Mankaran Cheema on August 20, 2013

In example 3 why did you write -1 when it was supposed to be 1 and the answer would be x is greater than or equal to -2 where as you wrote x is -3 over 2