Math101 · Independent Ontario learning library
Math101Inequality
An inequality compares values that need not be equal. Symbols $<$ and $>$ are strict; $\le$ and $\ge$ include equality.
Inequalities express bounds, budgets, tolerances, feasible regions, and comparisons. Their interval solutions describe ranges of possibilities rather than one exact state.
Intuition and core definition
An inequality compares values that need not be equal. Symbols $<$ and $>$ are strict; $\le$ and $\ge$ include equality. A solution to an inequality is any value making the comparison true, so solutions are often intervals rather than single numbers.
Notation, language, and conditions
$a<b$ means $a$ lies left of $b$ on a number line. A compound inequality such as $2<x\le5$ means both conditions hold. Interval notation writes this as $(2,5]$: a parenthesis excludes and a bracket includes. Multiplying or dividing both sides by a negative number reverses the comparison.
Why this idea matters
An inequality describes an ordered range rather than a single balance point, with endpoint inclusion communicated by the comparison symbol.
A dependable method
- Identify the quantities being compared and whether endpoints are included.
- Simplify both sides and isolate the variable using operations on both sides.
- Reverse the inequality only when multiplying or dividing by a negative value.
- Represent the solution with notation and a number line.
- Test one interior value and relevant boundary values in the original inequality.
