Key takeaway: An inequality compares two expressions that are not necessarily equal using the symbols <, >, ≤ or ≥, and its solutions are usually a whole range of numbers rather than one value.
Grade 66.EE.B.8Expressions and Equations
Definition of Inequality
An inequality says that one quantity is smaller than, larger than, or at least as large as another. Many rules are written this way: a roller coaster may require riders to be at least 48 inches tall, written \(h \geq\, 48\).
Unlike an equation, which usually has a single solution, an inequality has a solution set, often infinitely many numbers. Any height of 48 inches, 51.25 inches or 60 inches works, while 45.5 does not. The set is drawn on a number line: a closed dot when the endpoint is included, an open dot when it is not, and an arrow showing the direction the solutions continue.
Formula and notation
The four symbols are
\[a < b \quad a > b \quad a \le b \quad a \ge b\]
read “less than”, “greater than”, “less than or equal to” and “greater than or equal to”. The small end of the symbol points at the smaller number.
Key facts
- The symbols with a bar underneath include the endpoint; the plain ones do not.
- A closed dot on a graph means the endpoint is a solution, an open dot means it is not.
- Testing one number from each region shows which part of the line to shade.
- Real situations use words like at least, at most, no more than and more than.
Inequality example
A ride requires \(h \geq\, 48\) inches. Which heights are allowed: 45.5, 48, 51.25? Describe how to graph the rule.
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45.5 is less than 48, so it fails.
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48 equals the limit, and the symbol \(\geq\,\) includes equality, so it passes.
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51.25 is greater than 48, so it passes.
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Graph: closed dot at 48 and a shaded arrow to the right.
The heights 48 and 51.25 are allowed; 45.5 is not.
Common mistake
Watch out
Kira draws an open dot at 48 for \(h \geq\, 48\).
Why it is wrong
An open dot means 48 is excluded, which is the graph of \(h > 48\). Because the rule says “at least” and allows exactly 48 inches, the dot must be closed.
Related terms and lessons
- Grade 6 · Writing and Graphing InequalitiesLessonWorksheetQuiz
- Algebra 1 · Linear and Compound InequalitiesLessonWorksheetQuiz
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