Write each statement using an absolute value equation or inequality. p is at least 3 units from 1.

Verified step by step guidance

1. Identify the key phrase: "p is at least 3 units from 1." This means the distance between p and 1 is greater than or equal to 3.
2. Recall that the distance between two numbers p and 1 on the number line can be expressed using absolute value as \(|p - 1|\).
3. Translate "at least 3 units" into an inequality: the distance is greater than or equal to 3, so write \(|p - 1| \geq 3\).
4. Understand that this absolute value inequality means p is either 3 or more units to the right of 1, or 3 or more units to the left of 1.
5. Optionally, rewrite the inequality without absolute value as two separate inequalities: \(p - 1 \geq 3\) or \(p - 1 \leq -3\).

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative. This concept helps express distances regardless of direction.

Distance on the Number Line

Distance between two points on the number line is the absolute value of their difference. For points p and a, the distance is |p - a|. This allows us to translate verbal statements about distance into absolute value expressions.

Inequalities Involving Absolute Value

An inequality with absolute value, such as |x - a| ≥ b, describes all values of x whose distance from a is at least b units. This is used to model conditions like 'at least' or 'no less than' in terms of distance.