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 most 4 units from 2

Short answer

$|x-2|\le 4$

Step-by-step solution

Step 1. The inequalities for the respective phrases.

The inequality for the phrase all real numbers x at most a units from b is$|x-b|\le a$

Step 2. Expressing the phrase in an inequality.

The real numbers x which are at most 4 units from 2 are those between$2-4\mathrm{and}2+4$

$2-4\le x\le 2+4$

$2-4-2\le x-2\le 2+4-2$

$-4\le x-2\le 4$

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

Since,

$-4\le x-2\le 4$

So,

$|x-2|\le 4$