Absolute Value Inequalities

Two patterns, and it is worth memorising which is which.

Less than: |x| < 5 means x is within 5 of zero, so −5 < x < 5. One connected chunk. Sometimes called "less thAND."

Greater than: |x| > 5 means x is further than 5 from zero, so x < −5 or x > 5. Two separate pieces. "Greator."

Isolate the absolute value first, then apply the pattern.

Two shortcuts. If an absolute value is greater than a negative number, it is always true: every real number works. If it is less than a negative, no solution ever.

Reading |u| as distance makes both patterns immediate. |u| < a says u lies within a of 0, giving the conjunction −a < u < a. |u| > a says u lies outside that window, giving the disjunction u < −a or u > a. For a ≤ 0 the results degenerate: |u| < a is empty for a ≤ 0, and |u| > a is all reals for a < 0.

The general distance reading is more useful still. |x − c| < r describes the open interval of radius r centred at c, which is exactly the ε-δ notation of calculus and the tolerance notation of engineering: a specification of 5.00 ± 0.02 cm is the inequality |x − 5| ≤ 0.02.

Since both patterns produce compound inequalities, they inherit all of the rules from that topic — including reversing both signs when multiplying through by a negative.