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Interval Notation

Date: 11/01/97 at 14:34:24
From: Phil
Subject: Interval notation

Question: Solve for x and write the answer in interval notation:

-3 < -x < 2


I tried to find values of x that can fit the question. Ex: making -x 
equal 3, dividing it by 3, and getting 1. I am reading the statement 
from right to left as 2 is greater than say 1 is greater than -3.  
I get lost when I think I am done by saying my answer is [-3] and how 
to show my work or even figure the question out. 
Can you please explain where I should start?

Thank you!

Date: 11/01/97 at 22:18:13
From: Doctor Scott
Subject: Re: Interval notation

Hi Phil!

Sometimes it helps to think of problems like this as being two 
problems connected by "and."  

The original statement says that  -x/3 > -3 AND -x/3 < 2. These two 
inequalities are relatively easy to solve: just multiply both sides 
by 3 and divide by -1 (or multiply both sides by -3). Remember, 
though, that when you multiply (or divide) an inequality by a 
negative number, you must change the order of the inequality. So,

                   -x/3 > -3
                      x < 9    <--multiply by -3 and change inequality

And,               -x/2 < 2
                      x > -4   <--multiply by -3 and change inequality

So, x < 9 AND x > -4.  Another way to write this is to write 
9 > x > -4, which says the same thing. Or, in interval notation, 
(-4, 9), using parentheses because x is strictly less than 9 and 
strictly greater than -4, and not equal to either number.

Now, with that in mind, notice that we really just did the same thing 
in two inequalities. So, it is sometimes convenient to combine all of 
the work into one step:

             -3 < -x/3 < 2

           * -3     * -3   * -3   <--multiply all three parts by -3 to 
                                     isolate the "x" in the middle.
                9  > x  > -4      <--change inequality.

Then, since x is between -4 and 9, we can write (-4, 9) as our 
solution in interval notation.

-Doctor Scott,  The Math Forum
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Associated Topics:
High School Basic Algebra
Middle School Algebra

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