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Review

Now, before we get into how to graph two-variable inequalities, let’s go over how to simplify inequalities in general.

Inequalities are essentially the same as any other equation (even though they are not equations), we try to simplify them the best we can so we know what numbers we can plug into them to make them true.

So essentially we solve them the same way as we would solve an equation.

However, when you deal with inequalities with two variables, you treat them basically the same way you would treat a one variable inequality, by simplifying until you can get y by itself.

It’s basically the same way we try to simplify two variable equations.

This is one of those times where it is harder to explain rather than just show, so without further ado…..

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SOME EXAMPLES OF INEQUALITIES

So, instead of going through and explaining, it may be better to just start off on some examples:

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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ANOTHER EXAMPLE

Solve the inequality:

y - 3x < 6x + 9

+ 3x + 3x

And we would end up with:

y = 9x + 9

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 = 6x + 9

+ 3x + 3x

And we would end up with:

y < 9x + 9

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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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THIRD EXAMPLE

Solve the inequality:

y + 4x > 5x -10

- 4x - 4x

And we would end up with:

y = x - 10

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 + 4x = 5x -10

- 4x - 4x

And we would end up with:

y > x - 10

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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 HOW DO WE GRAPH AN INEQUALITY?

So how do we graph inequalities?

It’s way more simple than you may think.

First we graph what the equation would like.

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

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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.

And that’s all

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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.

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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.

And that’s all

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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.

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.

And that’s all

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Example:

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

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.

And that’s all

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

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Example:

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

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.

And that’s all

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