Write each statement using an absolute value equation or inequality.
q is no more than 8 units from 22.

Verified step by step guidance

1. Identify the key phrase "no more than 8 units from 22," which means the distance between q and 22 is at most 8.
2. Recall that the absolute value expression \(\left| q - a \right|\) represents the distance between q and a on the number line.
3. Translate the phrase into the inequality \(\left| q - 22 \right| \leq 8\), where the absolute value measures the distance from 22.
4. Understand that the inequality \(\left| q - 22 \right| \leq 8\) means q is within 8 units either less than or greater than 22.
5. This absolute value inequality correctly represents the statement "q is no more than 8 units from 22."

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 number x, |x| denotes this distance, so |x| ≥ 0. This concept helps express distances regardless of direction.

Translating Distance Statements into Absolute Value

When a variable is described as being within a certain distance from a number, it can be expressed as an absolute value inequality. For example, 'q is no more than 8 units from 22' translates to |q - 22| ≤ 8, representing all values q within 8 units of 22.

Formulating Absolute Value Inequalities

Absolute value inequalities like |x - a| ≤ b describe all values x within a distance b from a fixed point a. This inequality can be rewritten as a compound inequality a - b ≤ x ≤ a + b, which helps in solving or graphing the solution set.