Answer: Note:
We want all numbers $\,x\,$ whose distance from zero is $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is...}}{\strut \ \ \ =\ \ \ } \overset{\text{three}}{\strut 3} $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:

1)	We want numbers [beautiful math coming... please be patient] $\,x\,$...	(the unknown is $\,x\,$)
2)	whose distance from zero...	(the vertical bars, $|\ |$, ask for distance from zero)
3)	is...	(the equal sign)
4)	$3$	three

Remember that you can ‘walk’ from zero in two directions: to the right, and to the left.
The number [beautiful math coming... please be patient] $\,3\,$ is three units from zero to the right;
the number $\,-3\,$ is three units from zero to the left.


The word ‘or’ in the sentence ‘$\,x=3\text{ or }x=-3\,$’ is the mathematical word ‘or’.
You may want to review its meaning:   Practice with the Mathematical Words ‘and’, ‘or’, ‘is equivalent to’

Answer: Note:
We want all numbers $\,x\,$ whose distance from zero is less than $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is less than...}}{\strut \ \ \ \ \ \lt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ } $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:

1)	We want numbers [beautiful math coming... please be patient] $\,x\,$...	(the unknown is $\,x\,$)
2)	whose distance from zero...	(the vertical bars, $|\ |$, ask for distance from zero)
3)	is less than...	(the ‘less than’ symbol)
4)	$3$	three

You can walk less than three units to the right—this gets you the numbers from [beautiful math coming... please be patient] $\,0\,$ to $\,3\,$.
You can walk less than three units to the left—this gets you the numbers from $\,0\,$ to $\,-3\,$.
Together, you end up with all the numbers between $\,-3\,$ and $\,3\,$:


The sentence ‘$\,-3 \lt x \lt 3\,$’ is just a shorthand for ‘$\,-3\lt x\ \text{ and }\ x<3\,$’.
That's the mathematical word ‘and’.
You may want to review its meaning:   Practice with the Mathematical Words ‘and’, ‘or’, ‘is equivalent to’

The shorthand ‘$\,-3\lt x\lt 3\,$’ is a great shorthand, because you see an [beautiful math coming... please be patient] $\,x\,$ trapped between $\,-3\,$ and $\,3\,$;
and those are precisely the values of $\,x\,$ that make the sentence true.

Answer: Note:
We want all numbers $\,x\,$ whose distance from zero is greater than $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is greater than...}}{\strut \ \ \ \ \ \gt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ } $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:

1)	We want numbers [beautiful math coming... please be patient] $\,x\,$...	(the unknown is $\,x\,$)
2)	whose distance from zero...	(the vertical bars, [beautiful math coming... please be patient] $|\ |$, ask for distance from zero)
3)	is greater than...	(the ‘greater than’ symbol)
4)	$3$	three

You can walk more than three units to the right—this gets you all the numbers to the right of $\,3\,$.
You can walk more than three units to the left—this gets you all the numbers to the left of $\,-3\,$.
Together, you end up with the two pieces shown below:


The word ‘or’ in the sentence ‘$\,x\lt -3\ \text{ or }\ x\gt 3\,$’ is the mathematical word ‘or’.
You may want to review its meaning:   Practice with the Mathematical Words ‘and’, ‘or’, ‘is equivalent to’

Solution:
There are no solutions, since distance can't be negative.
That is, [beautiful math coming... please be patient] $\,|x|\,$ is always greater than or equal to zero; so, it can't ever be less than $\,-1\,$.

Solution:
All real numbers are solutions, since all distances are nonnegative.
That is, [beautiful math coming... please be patient] $\,|x|\,$ is always greater than or equal to zero; so, it must also be greater than $\,-1\,$.