Concept 4

Solve problems with absolute value, intervals, and inequality notation.

Translate absolute value inequalities into intervals and read the endpoint notation correctly.

Core Idea

Absolute value measures distance from 0, so solve it by turning it into a two-sided inequality and then match the endpoint symbols to interval notation.

Understanding

Rule: An absolute value inequality is usually easiest if you translate it into a range.

For $| 𝑒 𝑥 𝑝 𝑟 𝑒 𝑠 𝑠 𝑖 𝑜 𝑛 | < 𝑎$ , the expression stays between $- 𝑎$ and $𝑎$ . For $| 𝑒 𝑥 𝑝 𝑟 𝑒 𝑠 𝑠 𝑖 𝑜 𝑛 | \leq 𝑎$ , the endpoints are included. After solving, read the symbols carefully when writing interval notation: parentheses mean the endpoint is not included, and brackets mean it is.

Question

Worked Example

Which interval represents the solution to $| 2 𝑥 - 3 | \leq 7$ ?

Select an answer to see the explanation