inequalities

Question

So I was looking at my math book and it was describing inequalities and it was showing that |X| < r then translated that to

-r< x < r and I was confused on how they got -r out of nowhere.

Answer 1

Great question Camilo.  Short answer to your question is that's a rule when you're working on absolute values.  

If you can use a number line and pick a value for r (say 4), you'll see that to make |x|<r be true you must include all the values between -4 and 4.  In other words  -4 < x < 4

 

If you'd like to better understand why it's a rule, read on, and then you'll be able to do an example on your own.

 

For this kind of notation (that is, the symbols used to communicate the math), it's sometimes helpful to do two things:  1) Be sure to understand the symbols like |X|;  2) write down an easy example for yourself using a number line

 

"Absolute value" as you know means turn the value positive if it's negative... 

Example  |-4| is just 4,  and |-x| is x.   See?  you can remove the  | | sign when you change the negative to a positive.

 

In your book, |x|< r  means the absolute value of x has to be less than some value we call "r" (let's pick 4 for r in this case).

So for all positive values of x (including zero), you can just write x < 4.  Hey, we're half way done.

And for all negative values of x, you would need to use the rule of reversing the arrow while removing the | | symbol:

So write that as: x > -r

Now think about x being "greater" than -r... that means "more positive than" -r, and therefore the absolute value of x will be less than r as was required in the first place. 

 

Get it? Let me know!