Equation field guide
Inequalities: bounds and allowed ranges
An inequality compares values without claiming they are equal. It describes a range, threshold, or bound that a quantity may satisfy.
Reading the symbols
< means less than; > means greater than; ≤ and ≥ include equality. Thus x < 3 excludes 3, while x ≤ 3 includes it.
The direction matters. 3 < x says the same thing as x > 3, but x < 3 says the reverse. Reading the symbol aloud can prevent an accidental reversal.
Operations can change direction
Adding or subtracting the same quantity on both sides preserves an inequality. Multiplying or dividing by a positive number also preserves it.
Multiplying or dividing by a negative number reverses the direction: if −2x < 6, dividing by −2 gives x > −3. This reversal reflects the flipped order of negative numbers on a number line.
Bounds in technical writing
Inequalities commonly state safety limits, error bounds, confidence thresholds, domains, and optimization constraints. For example, 0 ≤ p ≤ 1 says a probability must lie between zero and one.
A strict bound can be important. “Less than” and “less than or equal to” differ exactly at the boundary value.