Algebra 1 Compound Inequalities

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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 conditions0230

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

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