Absolute Values and Inequality understanding

[jzapata87]

Homework Statement

I saw this in my real analysis textbook and I have been trying to understand how this equation $$\left | x-c \right |< 1$$

you can get this:$$\left | x \right |\leq \left | c \right | + 1$$

Homework Equations

I wanted to know what steps made this possible , particularly why it changed from $$< to \leq$$

The Attempt at a Solution

My thinking was that, they did this because it makes no difference as to if you put $$< or \leq$$

Any help is appreciated!
Thanks

 

[hikaru1221]

$$|x|-|c|\leq |x-c| < 1 \Rightarrow |x|-|c| \leq 1$$
This one is true because of the symbol $$\Rightarrow$$ (or as you said, it makes no difference). Things would be different if it were $$\Leftrightarrow$$.

 

[Susanne217]

$$|x|-|c|\leq |x-c| < 1 \Rightarrow |x|-|c| \leq 1$$
This one is true because of the symbol $$\Rightarrow$$ (or as you said, it makes no difference). Things would be different if it were $$\Leftrightarrow$$.

You forgot to tell the young man that to understand this problem he also needs to understand two of the three basic norms Homogeneity and Subadditivity