How do you find absolute value without using a graph?

Question

I am very confused on how to find the absolute value of an equation without graphing it first. Can you help me?

Answer 1

The definition of absolute value is |x| = { x if x >=0; or -x if x < 0 }.

Note that sqrt(x^2) = |x|.

 

E.g., if y = x^2 - 1, find |y|.

 

Case 1: If x^2 - 1 < 0, or x^2 < 1, or sqrt(x^2) < sqrt(1), or |x| < 1, or -1 < x < 1,

            then |y| = -y = 1 - x^2.

Case 2: If x^2 - 1 >= 0, or x <= -1 or x >= 1,

            then |y| = +y = x^2 - 1.

 

|y| is a piecewise function: |y| = { 1 - x^2 if -1 < x < 1; or x^2 -1 otherwise }.

Answer 2

 

l Q l  =  Q   for  Q >0

l  Q I   =  - Q   for  Q <0

 

 

    Example  :   Evaluate the following expression for   5 < X < 9 

 

    l  l X - 14 l - l x - 3  l  l  =

 

    l - X +14  + X - 3 l  = l 14-3 l = l 11l = 11

Answer 3

Dear Jaws,

 

Let's take an example:

 

Suppose you are given this inequality regarding absolute value

 

⌈x – 1⌉ < 3

 

All you have to do to get rid of the absolute value notation is to rewrite the expression as

 

-3 < (x - 1) < 3

 

Add 1 to each side to get

 

-2 < x < 4

 

It doesn't make any difference if you are dealing with =, <, >, ≤ or ≥, the idea is the same.  

 

In general terms, if f(x) is some function of x

 

⌈f(x)⌉ =, or <, or >, or ≤, or ≥ n, then

 

-n =, or <, or >, or ≤, or ≥ f(x) =, or <, or >, or ≤, or ≥ n