Question

Write each of the following sets by listing their elements between braces. $$ \\{x \in \mathbb{Z}:|2 x|<5\\} $$

Short answer

The elements of the set are \{ -2, -1, 0, 1, 2 \}.

Step-by-step solution

Understand the absolute value function

The absolute value of a number is its distance from zero, disregarding the direction. It is well-known that \( |a| < b \) if and only if \( -b < a < b \). In this case, \( |2x| < 5 \) is equivalent to \( -5 < 2x < 5 \).

Solve inequalities

To find the integral solutions for \( x \), the inequalities must be solved. Divide the entire inequality by 2: \( -2.5 < x < 2.5 \). Here, \( x \) must be an integer.

Identify the integers

Identify the integers falling within the range. Since \( x \) is defined as an element of the set of integers \( \mathbb{Z} \), solutions are \( x \) such that \( -2.5 < x < 2.5 \). The integers within this range are \( -2, -1, 0, 1, 2 \).