Algebra 1 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 x² = 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 × 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 I’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 there0329

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 x’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

I’m not going to focus a whole lot on that set builder notation but you will definitely see me use some inequalities, some number lines0397

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 x’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 I’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 = – 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 ÷ -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 to0793

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 painted0850

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 B0958

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

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