Absolute Value Inequalities Calculator

Math

This absolute value inequality calculator checks whether a number x satisfies |x| ≤ a, the most common form of absolute value inequality. Enter your value x and the bound a, and it tells you whether x lies within the interval −a ≤ x ≤ a. Geometrically, |x| ≤ a means x is no farther than a from zero on the number line.

Formula

|x| ≤ a means -a ≤ x ≤ a
  • The inequality |x| ≤ a says that x is at most a units away from 0 on the number line.
  • That is the same as the double inequality −a ≤ x ≤ a, so the solution set is the closed interval [−a, a].
  • The calculator compares |x| with the bound a and reports True when |x| ≤ a, otherwise False.
  • If the bound a is negative, no value of x can satisfy |x| ≤ a, so the result is always False.
  • For the strict version |x| < a, the endpoints −a and a are excluded from the interval.

Example Calculation

Inputs
  • Value: 5
  • Absolute Value Bound: 3

With x = 5 and a = 3, |5| = 5, which is greater than 3, so x does not satisfy |x| ≤ 3 and the result is False. A value like x = 2 would satisfy it, since |2| = 2 ≤ 3.

Frequently asked questions

What does |x| ≤ a mean?
It means x is within a distance of a from zero, which is the same as −a ≤ x ≤ a. The solution is the interval [−a, a].
How does this calculator work?
It takes the absolute value of your number and compares it with the bound a. If |x| ≤ a it returns True; otherwise it returns False.
What is the difference between |x| ≤ a and |x| ≥ a?
|x| ≤ a gives a single interval [−a, a] near zero. |x| ≥ a gives two pieces, x ≤ −a or x ≥ a, the values far from zero.
What if the bound a is negative?
Then |x| ≤ a has no solutions, because an absolute value cannot be less than a negative number, so the result is always False.
Does it handle strict inequalities like |x| < a?
This tool checks the non-strict form |x| ≤ a. For the strict form, the only difference is that the endpoints x = a and x = −a are not included.