Absolute Value Inequalities

Absolute value represents the distance a number lies from zero on the number line, regardless of whether the number is positive or negative. This distance is always expressed as a non-negative value, making it a fundamental tool for measuring magnitude without regard to direction or sign.

When the absolute value of a quantity is less than or equal to a given value, the solution includes all numbers between the negative and positive bounds. For example, |x| ≤ a means x must satisfy −a ≤ x ≤ a, creating a compound inequality that defines a continuous range of acceptable values.

If the absolute value is greater than or equal to a given value, the solution includes all numbers outside those bounds. For |x| ≥ a, the solution consists of two separate intervals: x ≤ −a or x ≥ a, representing values at a distance of at least a units from zero.

Solutions are shown on a number line using segments or rays, with closed circles indicating endpoints included because of the equality part of the inequality. For less-than inequalities, a solid line segment connects the bounds; for greater-than inequalities, rays extend outward from the endpoints.

Absolute value inequalities capture permissible deviations from an expected value in scientific contexts. For instance, measuring sound speed around 343 meters per second with a variation of ±2 m/s is modeled as |v − 343| ≤ 2, which defines the full range of acceptable measured values through mathematical modeling problem solving.

Solving involves subtracting and adding the allowed variation to the expected value to determine the full range of acceptable values. For |x − 80| ≤ 2.5, subtract and add 2.5 from 80 to get 77.5 ≤ x ≤ 82.5, representing all temperatures maintaining optimal reaction conditions.

Closed circles indicate that endpoints are included in the solution set because of the equality component of the inequality. When an absolute value inequality uses ≤ or ≥ symbols, the boundary values themselves satisfy the condition and must be marked as part of the solution on the number line.