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## How do you find absolute value without using a graph?

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

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 }.

If definition is given for l f(x) l = f(x)  if f(x) >0 , - f(x) if f(x)<0 is more generalized than defining only for x.

Example I have given below is a good example of evaluating absolute value function f(x)  for a defined domain of X.
x can represent the value of any number including the value of a function. x is a placeholder that can be replaced by any expression or function.

### 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

Note that l x- 14l  = - x + 14   for  5< X<9

l X - 3 l = X -3        for 5 <X< 9
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