Question

A phrase describing a set of real numbers is given. Express the phase as an inequality involving an absolute value.

All real numbers x at least 5 units from 7

Short answer

$|x-7|\ge 5$

Step-by-step solution

Step 1. The inequalities for the respective phrases.

The inequality for the phrase all real numbers x atleast a units from b is$|x-b|\ge a$

Step 2. Expressing the phrase in an inequality.

The real numbers x which are at least 5 units from 7 are those on the right of$7+5$ and those on the left of$7-5$

$\begin{array}{c}x>7+5\\ \mathrm{or},x-7>5\end{array}$

$\begin{array}{c}x<7-5\\ \mathrm{or},x-7<-5\end{array}$

Step 3. Expressing the phrase as an inequality involving an absolute value.

Since,

$x-7>5,x-7<-5$

So,

$|x-7|\ge 5$