SYSTEMS OF INEQUALITIES

OBJECTIVE

• Review over how to graph one variable inequalities
• Review over how to graph two variable inequalities
• Review over Sequences of Inequalities
• Do some examples
• Homework

Review over how to graph a one variable inequality

So, there are a few ways to graph an inequality.

When we are graphing one variable inequalities, we will always graph them on a number line.

Now the problem becomes how to graph it appropriately.

Direction of the arrow and placement of the circle

So, when it comes to graphing a one variable inequality, we need to make sure to place two things, a circle and an arrow.

Your circle goes over the number you found when you simplified the inequality, and the arrow goes in the direction that makes it true.

Here’s an example

Example

Simplify and graph the inequality:

2x + 3 > 23

- 3 - 3

2x > 20

_______� 2 2

x > 10

So first, we need to simplify the inequality

Now we graph it.�

So as we can see, we have: x > 10

This means we need to place our circle over 10 on our graph.

So here’s our graph

Now we need to place our circle on the graph over 10.

Now we need to place our arrow in the direction that makes this true.

Since x is greater than 10, this would mean all the numbers to the right would make it right.

So our graph should look like:

Open and Closed circle

So now there is one more rule when it come to graphing inequalities.

When you are graphing an inequality that is:

Greater than or equal to, or Less than or equal to, then your circle needs to be closed

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If you’re graphing an inequality that is just greater than, or less than, the circle needs to be opened.

EXAMPLE

Let’s say you have something like:

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x > 5

Now we have greater than, with no line underneath, so the graph for this would look like:

Now let’s say you have something else like:

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Now we have greater than or equal to, as you can see because there is a line underneath. �So the graph for this would look like:

So what about One Variable inequalities on an x-y coordinate plane?

So I know you guys were a little confused on how to graph one variable inequalities on an x-y coordinate plane (because IXL)

However, I want to go over it anyway just in case it happens again.

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How to graph one variable inequalities on an X-Y coordinate plane

First things first, we have to know what it is we are graphing, x, or y?

So, for example, consider the following problem:

X > 5

So, we know for a fact we are graphing just x, but what does that mean for y?

Well, to be blunt, it means we don’t care what y is because x is the only variable that is important.

So, since it’s the only variable that is important, we graph a vertical line on whatever x is on. (in this case, 5)

So, first thing we do is graph the line x = 5

Now, we know that we have greater than, not greater than or equal to, so let’s make this line dashed.

Lastly, we look and see that x much be bigger than 5, so we pick a number bigger than 5 on the graph, and shade that way.

That’s great and all, but now how do we graph just y instead?

Same way we graphed just x, with a little twist.

Here’s an example:

So, for example, consider the following problem:

y < 3

So, we know for a fact we are graphing just y, but what does that mean for x?

Well, to be blunt, it means we don’t care what x is because y is the only variable that is important.

So, since it’s the only variable that is important, we graph a horizontal line on whatever y is on. (in this case, 3)

So, first thing we do is graph the line y = 3

Now, we know that we have greater than, not greater than or equal to, so let’s make this line dashed.

Lastly, we look and see that x much be bigger than 5, so we pick a number bigger than 5 on the graph, and shade that way.

SO HOW DO WE GRAPH A TWO VARIABLE INEQUALITY?

Now, I’d like to review over how to graph a two variable inequality.

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Remember, first we graph what the equation would like.

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Next we shade the graph either up, or down depending on whether it is greater than or less than.

EXAMPLE:

Graph:

Y ≥ 3x + 5

First, we graph the line y = 3x + 5

Then we look and see that the inequality is great than or equal to, so we look at the line and shade up.

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And that’s all

SOME KEY POINTS

• Again, if the inequality is y is greater than or equal to, you graph the line and then shade up.
• If the inequality is y is less than or equal to, you graph the line and then shade down.

One last point to make:

If the graph is less than, or greater than, you do the same thing BUT MAKE SURE YOU MAKE YOUR LINE DASHED.

Example:

Graph:

Y > 3x + 5

First, we graph the line y = 3x + 5

Then we look and see that the inequality is great than, so we look at the line and shade up.

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And that’s all

Example:

Graph:

Y < 4x - 2

First, we graph the line y = 4x - 2

However, since this is just less than, we need to make sure to make our line dashed.

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Again, remember, no line, make it dashed.

Then we look and see that the inequality is less than, which means y needs to be less than the line.

So, this means we need to shade under the line.

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And that’s all

Example:

Now, since we can see that this is greater than or equal to, so we can keep the line solid.

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Again, remember, line, keep it solid.

Then we look and see that the inequality is greater than or equal to, which means y needs to be greater than the line.

So, this means we need to shade above the line.

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And that’s all

First, we graph the line y = 3x - 8

Example:

Now, since we can see that this is less than or equal to, so we can keep the line solid.

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Again, remember, line, keep it solid.

Then we look and see that the inequality is less than or equal to, which means y needs to be less than the line.

So, this means we need to shade under the line.

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And that’s all

First, we graph the line y = 5x + 6

Solving systems of Inequalities

Now that we’ve reviewed over how to graph multiple inequalities, it should be easier to graph a system.

It’s all been leading up to this!

So, imagine we are given a set of inequalities like:

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y > 3x -3

y < 2x + 2

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And they ask us to solve.

Well, this looks really complicated, but here’s how we do it.

First, pick an inequality

It really doesn’t matter which inequality you choose, but choose one.

After we choose our first inequality, we graph it.

So, let’s do:

y > 3x - 3

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

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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.

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Alright! We are half way done!

(Yes, I know, ugh)

Now we graph the other inequality

So, now we need to graph the other inequality.

There is a catch though, we need to make sure we graph the second inequality on the same graph.

So, let’s do:

y < 2x + 2

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

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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.

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Alright! Now let’s look at our graph to understand it.

UNDERSTANDING THE GRAPH

So let’s look at this graph.

Do you happen to see that the shaded regions from both inequalities seems to overlap itself?

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This is the answer we are looking for!

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The overlapped shaded parts are the answers to the inequality that works!

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So, just to make sure you understand, let’s do another.

Example 2

Graph the following Sequence of Inequalities:

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

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

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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.

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Now we graph the other one

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:

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

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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.

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

Those are the answers that these inequalities share.

Example 3

Graph the following Sequence of Inequalities:

y < 3x + 4

Again, first we pick an inequality.

Let’s do:

y < 3x + 4

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We know we need to first graph the equation: y = 3x + 4

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Now, since it’s less than, we know we need to make our line dashed.

�

So, now we look at the inequality.

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

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Now we graph the other one

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:

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

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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.

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

Those are the answers that these inequalities share.