Absolute Value Inequalities

The absolute value indicates the distance a number lies from zero on the number line, regardless of whether the number is positive or negative.

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.

If the absolute value is greater than or equal to a given value, the solution includes all numbers outside those bounds.

These solutions are shown on a number line using segments or rays, with circles indicating endpoints included because of the equality part of the inequality.

Consider measuring the speed of sound using echoes from a known distance to a wall. Depending on environmental fluctuations, the measured speed is around 343 meters per second, with a variation of plus or minus 2 meters per second.

This variation is modeled using an absolute value inequality, which sets a limit on how far the observed speed can deviate from the expected value.

Solving the inequality involves subtracting and adding the allowed variation to the expected speed to get the full range of acceptable values.