Compound Inequalities

Date: 09/10/97 at 07:06:11
From: patricia kalaw
Subject: Compound inequalities

1. Can you write compound inequalities that use "or" the same way 
   you write the compound inequalities that use "and"? e.g.

   x>3 and x<12 = 3<x<12
   x>3 or x<12 =(?) 3<x<12

2. How do I graph this?

Thanks!      

Date: 09/21/97 at 13:50:45
From: Doctor Chita
Subject: Re: Compound inequalities

Hi Patricia:

The first of the inequalities you've written is correct. The second is 
an incorrect use of the word "or" since the relation you want is 
unchanged.

When writing inequalities, an "or" signifies that there are two sets 
of numbers that do not intersect. In real life, it would be like males 
and females. One is either a male OR a female; not a male AND a 
female.

In math, "and" statements can be written in one block, such as 
2 < x < 10. This says that 2 is less than x and less than 10. 
x is a number between the two.

"Or" statements are usually written in two blocks, such as 
x < 2 or x > 10.  This means that x is less than 2 or greater than 10. 
(Notice that the inequality signs point in different directions.)  
The reason for the difference is that a number can't be in two places 
at once.  

In the first inequality I wrote, if the number is 5, it is between 2 
and 10, so it is BOTH larger than 2 and less than 10.

In the second inequality, if the number is 100, it is either less 
than 2 OR greater than 10. It would be impossible for the number 100 
to be less than 2 AND greater than 10. The break between the two 
blocks in an "or" statement implies that a number x is on one side of 
a boundary number or the other side of the other boundary number, not 
between the two boundary numbers as in an "and" statement.

The graphs reflect the way you write the inequality. If it's an "and" 
statement, shade the part of the number line between the two boundary 
numbers.

If it's an "or" shade the rays on either side of the two boundary 
numbers.
 
Put a circle around the boundary number(s) if there is no equality (=) 
involved. Fill the circle(s) if there is an equality.


Here is 2 < x < 10: x lies between 2 and 10.
 <----------o============o------------>
   2   10


Here is x < 2 or x > 10: x lies to the left of 2 or to the right 
of 10.

<===========o-------------o============>
   2     10


The secret is translating the symbolic statement into English, and 
understanding that you are trying to locate a number either between 
two given numbers or on either side of the given numbers. 

Got it?

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/