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Graphing Inequalities

Graphing Inequalities

Tutorials you should read before this one:  Graphing Real Numbers on a Number Line.

A brief explanation
Up until now, we have only had to deal with the equality symbol (=).  But what happens when two things are not equal? They can still be described in relationship to one another (compared), and to do that, we use inequality symbols.

Graphing Inequalities Basics
There are five basic inequality symbols:  < , <=, > , >= and NOT =
When graphing an inequality, aside from drawing the number line itself, there are three steps you should follow.  
  1. Read your inequality (correctly!).  
  2. Decide whether to use an open (O) or closed circle (*).  
  3. Finally, figure out which direction you need to shade in.
Reading Inequalities
When I introduced the five basic inequality symbols, notice that I refrained from labeling them.  The reason for that is because most students are taught that this symbol (>) automatically means "greater than."  This instant labeling of a symbol gets confusing when problems are not written in the normal direction, such as 4 > x. To always graph correctly, you must first read your inequality starting from the variable.  The good news is you can still use the old "Alligator Mouth" memory trick.  If the mouth is facing your variable, it's greater than.  If the tail is facing your variable, it's less than. But only if you always read beginning with the variable!  You will understand what I mean after a couple of examples.  

Open and Closed Circles
If your symbol has the "equal to option," which is that single line underneath the greater than/less than symbol, then you use a closed circle.  The closed circle is only used for the greater than or equal to or less than or equal to symbols.  
The open circle is used for >, < and not equal to

Shading your number line
Less than shades to the Left (they both start with L) because the "lesser" numbers are to the left.  Greater than shades to the right, because the further right you go, the "greater" the numbers become.  
Sometimes students think that they can always use this memory trick:   since inequality symbols look like the arrow ends on a number line, they think they can always shade towards the number line arrow that looks like the inequality symbol.   (If you don't know what I'm talking about, good!).  DON'T DO THIS.  Again, it will get you into trouble for those backwards problems.  

Step-by-Step Examples
x > -2We read this inequality as x is "greater than" negative 2.  (The mouth is facing the x).
open circle on -2Since our inequality does NOT have the equal to option, we use an open circle.
x > -2 shades to rightSince our inequality reads as "greater than," we shade to the right.
Always read your inequality starting from the variable.
>=  xThis reads as "x is less than or equal to 4" because this time, our tail is facing our variable.
5 >= xOur inequality symbol does have the equal to option, so we use a closed circle.
x less than equal to 5 shades to leftLess than always shades to the left.

-0.3 < x x is greater than  -0.362
open circle between 0 and -1-0.362 lies between 0 and -1, open circle.
shade to rightGreater than shades to right.

x < -7/2This time we have a fraction. Our inequality reads as "x is less than -7/2."
Remember, convert your fraction to its decimal equivalent = -3.5 (Divide -7 by 2).
-3.5 lies between -3 and -4 on our number line.
open circle between -3 and -4Open circle since our inequality does NOT have the equal to option.
Shaded to leftLess than shades to left.

x Not equal to -1This reads as x is not equal to -1.  
Open circle -1Open circle since -1 is our end number.
Shade both directionsThis time, we shade both directions, because all the numbers that are not equal to -1 are to the left (where the numbers that are less than -1 are located) and to the right (numbers that are greater than -1).