Lesson 9.3

Systems Of Inequalities

Some examples of inequalities

• 2X<Y
• 8X+5>Y
• 3X-9Y<48
• X ²+Y ²>7
• X>Y 

 Remember, the MOST IMPORTANT thing to remember when solving inequalities is that you are looking for an ordered pair that makes the statement true. 

Two inequalities are equivalent if they have the same solution. 

The graph of an inequality is the set of all points that correspond to the solution. 

To Solve an inequality we replace the inequality sign with an equal sign.  This divides the graph.  Every point on one side of the line will be true, while every point on the other side will  be false.  In order to determine which side is true we plug in points.

For example

 

The shaded region represents every value that makes the statement true.

Now, sketching the graph of Y<X+1

1. First, replace the < symbol with a = sign.  Y=X+1
2. Next, sketch the line

 

      3.  Then, plug in points.  We will use (2,0) and (0,2)

                    Since 0 <2+1 is true and 2< 0+1 is false we know to shade in all points below the line.  

      4.  Shade in. 

 

You can do the same thing for parabolas.  

Now it gets really fun.  

Solving a system of inequalities

The only thing that is different here is that we have two or more lines.  

Now, the solution will have to correspond to both of the inequalities. 

If we have the two inequalities X+2Y>-4 and 3X+Y<3 we want to find the area that should be shaded in. 

Our steps are just the same for this one.

1. First substitute in an = sign
2. then draw the line
3. next plug in points to see where the statement is true
4. and finally shade

 The area of overlap is the area we are looking for 

The graph will look something like this. 

This is a cool looking one. 

 Now, absolute values.  

• When given /X/ < b we know that -b< X <b
• And when given /X/ >b we know that X <-b or X > b  

 So, when given /X/ < 4 we know that we must shade the region between -4 and 4.

Here, /X/<3.5 and 0<Y<3.5

Systems of inequalities relating to a circle.

X ² + Y² = r<sup>2

 If</sup> 
     X ² + Y^{2  > 4} 
 
We shade outside of the circle

If

 X ² + Y^{2  < 4}

We shade inside the circle 

Ignoring the triangle this is what it looks like with <

When trying to find a system of inequalities form a graph it is helpful to remember the equation of a circle and slope intercept form.  And it is necessary to take note of which region is shaded, and where it is in relation to ALL of the functions and circles.  

And that's all you need to know about systems of inequalities.

Matthew "Seaglass" Silbergleit