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Review of how to simplify two variable inequalities

So, I recognize that we have been going over how to graph inequalities for quite some time, but we haven’t really touched on how to simplify them.

So, here is a quick refresher on how to simplify two variable inequalities.

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EXAMPLE 1: SIMPLIFY THE INEQUALITY

As a review, imagine you are given something like….

Solve the inequality:

y – 2 > 3x + 4

So, when you want to solve an inequality, the first thing you do is treat it like an equation.

So, if we were to have:

y – 2 = 3x + 4

+2 +2 And we get:

y = 3x + 6

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WE DO THE SAME THING TO THE INEQUALITY

So, if what they gave us is:

y – 2 > 3x + 4

+2 + 2

And so we are left with:

y > 3x + 6

So what does this mean?

This means that whatever number we pick for x, y must be greater than that number, times 3 plus 6.

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SECOND EXAMPLE

Solve the inequality:

y + 2 < 6x + 9

- 2 - 2

And we would end up with:

y = 6x + 7

Again we do the same thing with the inequality

Again, first thing we do is treat this like an equation.

So, if we had:

y + 2 = 6x + 9

- 2 - 2

And we would end up with:

y < 6x + 7

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LAST EXAMPLE

Solve the inequality:

y + 3x + 7 < 6x + 9

- 3x - 7 -3x - 7

And we would end up with:

y = 3x + 2

Again we do the same thing with the inequality

Again, first thing we do is treat this like an equation.

So, if we had:

y + 3x + 7 = 6x + 9

- 3x -7 - 3x - 7

And we would end up with:

y < 3x + 2

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Important note!

However, don’t forget the one rule that you need to be aware of.

If you divide (or multiply) an inequality by -1, you need to switch the inequality.

What I mean by that is this:

Example:

-2y < 4x + 10

We know we divide by -2 to get y by itself, however our answer would be:

y > -2x – 5

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So, now that we understand how to simplify, let’s review rearranging

Now, this may seem different, but the reality is a little refresher isn’t going to hurt.

I promise there is a method to my madness, but in the mean time let’s get started.

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So what do we mean by “rearranging equations”?

So what rearranging equations means, is switching the variables around a little to help us find what we’re looking for.

But in order to do this, we need to remember something…..�What is a variable?

A variable is just a number that we don’t know yet right?

So, since a variable is just a number that we don’t know yet, then maybe can we perform operations (+, -, *, /) on it like we would any other number?

Of course we can!

In fact, by either applying operations, or undoing operations, we can solve for things that we wouldn’t be able to otherwise!

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So, what does that look like?

Well, imagine you are given something like this:

x + y = 2

-y -y

x = 2 - y

And you’re asked to solve for x.

How would you do it?

Well, again, a variable is just a number we don’t know yet.

So, what if we subtracted y from both sides?

Now can we simplify this anymore?

Not really.

There’s no way for us to subtract y – 2 since we don’t know what y is.

So, our equation is:

x = 2 – y

Now, this was a simple case. Let’s try one a little harder.

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Example 2 (a little harder)

Now imagine you are given something like this:

3x + 6y = 18

-6y -6y

3x = 18 - 6y

And you’re asked to solve for x.

How would you do it?

Well, as we saw before, we can change any number in the equation, whether it’s a variable or not.

So, what if we subtracted 6y from both sides?

Now can we simplify this anymore?

Sure! We have 3x = 18 – 6y, what if we divide by 3 on both sides?

__ __ __

3 3 3

x = 6 – 2y

Then, our equation is:

x = 6 – 2y

This is essentially what you can expect, but just in case…

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WHAT IF THEY DON’T DIVIDE NICELY?

Sometimes, that happens, and we are left with fractions.

Here’s an example:

Imagine you are given something like this:

3y – 2x = 13

+ 2x +2x

3y = 2x + 13

And you’re asked to solve for y (just to change things up).

How would you do it?

Well, since we want to solve for y, and we have an x next y, why don’t we undo the x?

So, let’s add 2x to both sides!

Now can we simplify this anymore?

Well, there’s a 3 next to y, and we know when we have a number that is next to something that isn’t a number, it means multiplication.

So, let’s undo the multiplication by dividing by 3.

__ __ __

3 3 3

Now, our equation becomes:

So now we have:

It’s not pretty, but it does happen more frequently.

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THAT’S GREAT AND ALL, BUT CAN WE GET TO THE POINT NOW?

Alright, I know you have been patiently waiting for us to get to the point, so here it is.

Since we know we simplify inequalities the same way we simplify equations, then that also means that we can rearrange inequalities the same way we rearrange equations.

This actually helps us a lot when we need to solve a sequences of inequalities, especially when none of the inequalities are nice enough to be in slope intercept form.

Here’s an example.

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Example 1

Simplify the inequality:

-2x + 3y > x + 9

Again, we’re going to simplify this the same way we would an equation.

Now, because it is an inequality, we really want to simplify the inequality until we get y by itself.

This will help us graph it if we need.

So, to get y by itself, first we need to…..

+ 2x + 2x

And we get:

3y > 3x + 9

Now, we need to:

__ ___ __

3 3 3

And finally, we are left with:

y > x + 3

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But, wait, this looks familiar….

Didn’t we do this before?

Well, yes we did, but as more of a mention, not really as a lesson.

We went over rearranging an inequality to get y by itself, but you were never really told why to do that.

Just that you needed to.

However, now that we are tossing in a sequence of inequalities, you can now see why we make sure to solve for y.

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Example 2: Sequence example

Imagine you have the following:

Graph the following sequence:

3x + 3y > 12x – 9

y < 2x + 2

So, what should you do first?

Well, I don’t know about you, but I have no idea what the first inequality’s graph looks like.

So, how about we simplify it to slope intercept form first, then we can figure that out?

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So, let’s simplify

Again, we were given:

3x + 3y > 12x – 9

So let’s simplify!

First, since we are solving for y, let’s subtract 3x from both sides:

– 3x –3x

And we are left with:

3y > 9x – 9

Now, we divide by 3 on both sides

__ __ __

3 3 3

And finally we are left with:

y > 3x -3

So, now that we simplified the inequality, we can actually graph the sequence and see what answers the two share.

So, with the simplified inequality, we have:

y > 3x -3

y < 2x + 2

Again, first step is to choose one.

Let’s pick the one we just simplified:

y > 3x - 3

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Graphing our simplified inequality

Now that we’ve simplified it, let’s graph it.

So, again, our inequality was:

y > 3x - 3

First thing we need to do is graph the line: y = 3x - 3

Next, we notice that the inequality is greater than, not greater than or equal to, so we need to make sure our line is dashed.

Now, we need to shade.

Since y is on the left side of the inequality, and it is greater than, then we need to shade up.

Alright! We are half way done!

(Yes, I know, ugh)

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Now we graph the other inequality

So, now we need to graph the other inequality.

And of course, we need to make sure we graph the second inequality on the same graph.

So, let’s do:

y < 2x + 2

First thing we need to do is graph the line: y = 2x + 2

Next, we notice that the inequality is less than, not less than or equal to, so we need to make sure our line is dashed.

Now, we need to shade.

Since y is on the left side of the inequality, and it is less than, then we need to shade down.

And that’s all! �Let’s do another just in case though

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Example 3

Imagine you have the following:

Graph the following sequence:

Again, it looks like we have a very ugly looking inequality.

And, again, it’s next to impossible to graph that.

So, how about we simplify it first, then graph it?

At least we’ll get the right answer (and an A on the test!)

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So, let’s simplify

Again, we were given:

So let’s simplify!

First, since we are solving for y, let’s add 9 from both sides:

+9 + 9

And we are left with:

Now, we add 2x on both sides

+2x + 2x

And we are left with:

So, now that we simplified the inequality, we can actually graph the sequence and see what answers the two share.

Now, we divide by -5 on both sides

__ ____ ___

-5 -5 -5

Now, let’s make sure to switch the sign since we divided by a negative.

And we are left with:

So, with the simplified inequality, we have:

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Now let’s graph

So, to recap, we have:

We know we need to first graph the equation: y = 7x – 6

Now, since it’s less than or equal too, we know we need to keep the line solid.

�

So, now we look at the inequality.

Since y is on the left, and it’s less than or equal to, we know we need to shade down.

Now we graph the other one

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Now we graph the other inequality

So, now we need to graph the other inequality.

Again, remember we need to graph both on the same graph

So, let’s do:

First thing we need to do is graph the line: y = 4x - 6

Next, we notice that the inequality is greater than or equal to, so we can keep our line solid.

Now, we need to shade.

Since y is on the left side of the inequality, and it is greater than or equal to, then we need to shade up.

Again, the answer we are looking for is the shared shaded region.

Those are the answers that these inequalities share.

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And that’s all there is to it!

Essentially that is the lesson.

If you run into a sequence of inequalities that isn’t in slope-intercept form, then rearrange the inequality until it is.

Then keep doing what we’ve been doing!