Math101 · Independent Ontario learning library
Math101learn.math101.caInequality
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.
Worked example
Representations and interpretation
On a number line, a filled point at $-4$ and shading right represent $x\ge-4$. The same set appears as interval $[-4,\infty)$ and as all inputs where the graph of $y=-3x+4$ lies at or below $16$.
Reasoning about variations
Adding or subtracting any real number preserves order, as does multiplying by a positive number. A negative scale reflects the number line, which explains the reversal: from $2<5$, multiplying by $-1$ gives $-2>-5$.
Common mistakes
How to check your work
- Substitute the boundary and one point from each side into the original inequality.
- Translate the interval back into words and confirm endpoint inclusion.
- Graph both side-expressions and compare their vertical order.
Practice
- Solve $5x-7<18$.
- Write $-2\le x<4$ in interval notation.
- Is $3$ a solution of $2x+1\ge8$?
Answers and brief solutions
Show answers
- $x<5$ $5x<25$, so $x<5$.
- $[-2,4)$ $-2$ is included and $4$ is excluded.
- No $2(3)+1=7$, which is not at least $8$.
Synthesis and transfer
A theatre capacity rule becomes an upper-bound inequality; integer context, rather than algebra alone, determines the greatest allowable number of complete groups.
If a room holds at most $180$ people and each supervised group contains $14$ students plus one leader, the model $15g\le180$ gives $g\le12$. The whole-number condition comes from counting complete groups, not from changing the inequality rules. “At most” includes the capacity endpoint, while “fewer than” would exclude it. Testing $g=12$ and the next integer $g=13$ shows both that the proposed maximum works and that it cannot be increased. A graph on the real number line communicates the algebraic solution, but the contextual solution is a discrete subset. Separating those stages prevents careless rounding from admitting an infeasible count.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve $5x-7<18$.
- $5x<25$, so $x<5$.
End of lesson
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