Absolute Value Inequalities

⟺ ≤ x ≤

Tap a term to see what it means, or .

Calculate

Example: a greenhouse's temperature must stay within 1.5 °C of the ideal 22 °C: |x − 22| ≤ 1.5. Which temperatures are accepted? Switch to “at least” to see the opposite case. Change the values to work out your own case.

Centre a Radius r Distance

Step by step

What each term means

Absolute Value Inequality: |x − a| is the distance between x and a on the line, without sign. Asking for it to be at most r is asking for the x close to a.
Lower Bound (a − r): The leftmost point still within distance r of the centre.
Upper Bound (a + r): The rightmost point. With ≥ instead of ≤, the solution becomes the outside: x ≤ a − r or x ≥ a + r.