Inequalities
Mathematical inequalities are expressions that establish an order relationship between two values or algebraic expressions. Unlike equations, where both sides are equal, inequalities indicate that one value is greater, smaller, or different from another.
Notation
- \( a < b \rightarrow a\) is less than \(b\)
- \( a > b \rightarrow a\) is greater than \(b\)
- \( a \leq b \rightarrow a\) is less than or equal to \(b\)
- \( a \geq b \rightarrow a\) is greater than or equal to \(b\)
- \( a \neq b \rightarrow a\) is different from \(b\)
Visual guide
The difference between the greater than \((>\)) and less than \((<)\) signs is that they indicate opposite directions in a comparison of numerical values or algebraic expressions.
Simple explanation
- "Greater than" \((>)\) means that the number on the left is larger than the one on the right. Example: \( 5 > 2 \rightarrow\) "Five is greater than two."
- "Less than" \((<)\) means that the number on the left is smaller than the one on the right. Example: \(3 < 7 \rightarrow\) "Three is less than seven."
Visual trick
The inequality sign looks like an open mouth. Imagine that the mouth "wants to eat" the bigger number.
- In \(5 > 2\), the mouth opens toward \(5\) because it is the larger number.
- In \(3 < 7\), the mouth opens toward \(7\) because it is the larger number.
Graphical representation
| notation | inequality | graph |
|---|---|---|
| $$(a,b)$$ | $$a < x < b$$ | |
| $$[a,b]$$ | $$a \leq x \leq b$$ | |
| $$[a,b)$$ | $$a \leq x < b$$ | |
| $$(a,b]$$ | $$a < x \leq b$$ | |
| $$(a,\infty)$$ | $$a < x$$ | |
| $$[a,\infty)$$ | $$a \leq x$$ | |
| $$(-\infty,b)$$ | $$x < b$$ | |
| $$(-\infty,b]$$ | $$x \leq b$$ | |
| $$(-\infty,\infty)$$ | $$\infty \leq x \leq \infty$$ |
Fundamental properties
| property | example |
|---|---|
|
If \(a < b\) and \(b < c\), then \(a < c\). |
\(3 < 6\) and \(6 < 8\), so \(3 < 9\) |
|
If \(a < b\), then \( a + c < b + c \) and \( a - c < b - c \). |
\(2 < 5 \), so \( 2 + 2 < 5 + 2\) and \( 2 - 2 < 5 - 2 \) |
|
If \( a < b \) and \( 0 < c \), then \(ac < bc\) and \( \frac{a}{c} < \frac{b}{c} \). |
\(2 < 5 \) and \(0 < 3\), so \( 2 \times 3 < 5 \times 3\) and \(\frac{2}{3} < \frac{5}{3}\) |
|
If \(a < b\) and \( c < 0\), then \(ac \color{red}{>}\)\(bc\) and \( \frac{a}{c} \color{red}{>}\)\( \frac{b}{c}\). |
\( 2 < 5 \) and \(-3 < 0\), so \(2(-3) \color{red}{>}\)\( 5(-3)\) and \(\frac{2}{-3} \color{red}{>}\)\( \frac{5}{-3}\) |
Test
Instructions
Move the point along the number line until it matches the inequality displayed in the top left corner. To change the type of interval, click on the circle to toggle between an open or closed interval. Depending on the inequality, you can hide the arrows on the graph using the corresponding switches. Once you're sure of your answer, click the "Check" button. Your results and recommendations will appear at the bottom.