# How do you write: the sum of number and three no more than eight or is more than twelve of?

Oct 25, 2016

$n \le 5 \textcolor{w h i t e}{\text{XX")orcolor(white)("XX}} n > 9$
$\textcolor{w h i t e}{\text{XXX}}$or
$n \cancel{\in} \left(5 , 9\right]$

#### Explanation:

Let the number be represented by $n$

sum of number and three:
#color(white)("XXX")n+3

(sum of number and three) no more than eight:
$\textcolor{w h i t e}{\text{XXX}} n + 3 \le 8$

Here is where things get a bit unclear:
- is it the "number plus three" that is more than twelve?
or just "the number" that is no more than twelve.
- what is the significance of "of" in "twelve of"?
I will interpret this as:
$\textcolor{w h i t e}{\text{XXX}} n + 3 > 12$

So we have as a combination:
(sum of number and three no more than eight)
or
(sum of number and three more than twelve)
$\textcolor{w h i t e}{\text{XXX")n+3 <=8color(white)("XX")orcolor(white)("XX}} n + 3 > 12$

This could be simplified to
$\textcolor{w h i t e}{\text{XXX")n <= 5color(white)("XX")orcolor(white)("XX}} n > 9$

Note that this can not be written as a single algebraic inequality
but it could be written using a set of ranges:
$\textcolor{w h i t e}{\text{XXX}} n \in \left\{\left(- \infty , 5\right] , \left(9 , + \infty\right)\right\}$
or
$\textcolor{w h i t e}{\text{XXX}} n \cancel{\in} \left(5 , 9\right]$