Graphing systems of Inequalities, Part One.

Graph the solution set of the system of inequalities.

Here I have two linear inequalities.

In order to graph the system, I'm going to

graph each linear inequality separately on

the same coordinate plane and then shade the

overlap of the inequalities.

So first, we're going to take the top inequality,

which is 3x minus 4y is less than or equal

to 12, and I want to find the x and y-intercept

in order to graph the line.

The x-intercept is found by substituting a

0 in for y and I want to change the inequality

sign to an equal sign.

So, I have 3x minus 4 times 0 equals 12.

That's 3x minus 0 equals 12, which gives me

3x equals 12, and then, when we divide both

sides by 3, I get x equals 4.

So, my x-intercept is (4,0).

Next, we're going to find our y-intercept

and the y-intercept is found by substituting

in a 0 for x.

So, I have 3 times 0 minus 4y equals 12.

This gives me 0 minus 4y equals 12, which

is a negative 4y equals 12 and I'll divide

both sides by negative 4 and I get y equals

a negative 3.

So, my y-intercept is (0,-3).

Now we'll take these two points and plot them

on the graph: (4,0) is here and then we have

(0,-3), which is here.

If we look at my inequality sign, I have an

equal to line underneath the sign.

So, that means that my line is going to be

solid.

Now I want to decide where to shade on the

graph.

Do I shade above the line or below the line?

In order to determine this, we're going to

choose a test point and we can choose whatever

point we want as long as it's not on the line.

I'm going to choose a test point of (0,0)

and we'll plug this into the first inequality.

So, I have 3 times 0 minus 4 times 0 is less

than or equal to 12.

0 minus 0 is less than or equal to 12, so

0 is less than or equal to 12.

That is a true statement, so that means that

I'm going to shade above the line to include

that (0,0) point.

So, all this will be shaded.

That's the graph of my first inequality.

Now I'm going to graph the second inequality.

We'll look at our second inequality and first

you'll find the x-intercept.

So, I'll plug a 0 in for y and I get 4x minus

0 equals 8.

So, 4x equals 8.

Then, we'll divide both sides by 4 and we

get x equals 2.

So, my x-intercept is (2,0).

Next, we'll find the y-intercept.

We'll plug in a 0 for x and solve for y.

So, I have 4 times 0 minus 2y equals 8.

0 minus 2y equals 8 is negative 2y equals

8.

We'll divide both sides by negative 2 and

we get y equals negative 4.

So, my y-intercept is (0, -4).

Now I'm going to take both of these points

and plot them on the same graph that I plotted

the first inequality.

So, we'll find (2,0), which is here, and then

(0,-4), which is here.

We'll go to our inequality and look at the

sign.

We do not have an equal to line underneath

the sign, so I know my line is going to be

dashed or dotted.

Now I want to decide what side of the line

to shade.

We can choose whatever point we want as long

as it's not on the line.

I'm going to choose the point (1,2), which

is there.

So, I plugged in a 1 for x and a 2 for y into

the original inequality and we'll solve.

So, we get 4 minus 4 is greater than 8 and

then we have 0 is greater than 8.

That is not a true statement.

That means the point (1,2) is not going to

be included in my shading.

That means that I'll shade on the other side

of the line, which is over here.

Now, the solution to my equation is the graph

of the two lines plus the shading of the overlap.

So, I only want to include the shading of

the overlap of the two lines and as you can

see the overlap is going to be within these

two lines here.

So, my solution is what you see in the graph.

I have my first equation with a solid line,

that's a black line, and then I have my second

equation with a dotted line, that's the blue

line, and then the overlap of the shadings

is represented by the purple shading, which

is in the top middle of the two lines.