Question

Explain why \(|x-5|<2\) means that \(x-5\) is between \(-2\) and 2.

Short answer

The given inequality \(|x-5|<2\) implies that the distance of \(x\) from 5 on the number line is less than 2 units. This translates to the range of values 3 < \(x\) < 7. Therefore, the statement \(x-5\) is between -2 and 2 is true because \(x-5\) represents a distance less than 2 units from either side of 5, which means \(x\) is between 3 and 7.

Step-by-step solution

Understanding the Absolute Value

First, understand what the absolute value, denoted as \(|x|\), means. The absolute value of a number is its distance from zero on the number line, ignoring direction. Hence, \(|x-5|\) represents the distance of \(x\) from 5 on the number line.

Understanding the Inequality

Next, evaluate the inequality \(|x-5|<2\). This implies that the distance of \(x\) from 5 on the number line is less than 2 units. This can occur in two situations: when \(x\) is less than 5 by less than 2 units and when \(x\) is more than 5 by less than 2 units.

Translate into a Range of Values

Now, interpret the inequality into a range of \(x\) values. The distance of \(x\) from 5 being less than 2 units translates to \(x\) being more than 5-2=3 and less than 5+2=7. Hence, \(x\) lies between 3 and 7, and this is the interpretation of the inequality \(|x-5|<2\).