# How do you write the inequality and solve "eight less than a number is no more than 14 and no less than 5#?

Apr 14, 2017

See the entire solution process below:

#### Explanation:

First, let's call "a number": $n$

Then. "eight less than a number" can be written as:

$n - 8$

This expression "is no more than 14" can be written as:

$n - 8 \le 14$

This expression "and no less than 5" can be written as:

$n - 8 \ge 5$

Or

$5 \le n - 8$

We can combine these two inequalities as:

$5 \le n - 8 \le 14$

To solve we can add $\textcolor{red}{8}$ to each segment of the inequality to solve for $n$ while keeping the system of inequalities balanced:

$5 + \textcolor{red}{8} \le n - 8 + \textcolor{red}{8} \le 14 + \textcolor{red}{8}$

$13 \le n - 0 \le 22$

$13 \le n \le 22$

Or

$n \ge 13$ and $n \le 22$

Or

$\left[13 , 22\right]$