Key Concepts
Inequality Signs
The box below shows the symbol, meaning, and an example for each inequality sign.
| Symbol | Words | Example |
|---|---|---|
| [latex]\neq [/latex] | not equal to | [latex]{2}\neq{8}[/latex], 2 is not equal to 8 |
| [latex]\gt[/latex] | greater than | [latex]{5}\gt{1}[/latex], 5 is greater than 1 |
| [latex]\lt[/latex] | less than | [latex]{2}\lt{11}[/latex], 2 is less than 11 |
| [latex] \geq [/latex] | greater than or equal to | [latex]{4}\geq{ 4}[/latex], 4 is greater than or equal to 4 |
| [latex]\leq [/latex] | less than or equal to | [latex]{7}\leq{9}[/latex], 7 is less than or equal to 9 |
The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.
| Inequality | Words | Interval Notation |
|---|---|---|
| [latex]{a}\lt{x}\lt{ b}[/latex] | all real numbers between a and b, not including a and b | [latex]\left(a,b\right)[/latex] |
| [latex]{x}\gt{a}[/latex] | All real numbers greater than a, but not including a | [latex]\left(a,\infty \right)[/latex] |
| [latex]{x}\lt{b}[/latex] | All real numbers less than b, but not including b | [latex]\left(-\infty ,b\right)[/latex] |
| [latex]{x}\ge{a}[/latex] | All real numbers greater than a, including a | [latex]\left[a,\infty \right)[/latex] |
| [latex]{x}\le{b}[/latex] | All real numbers less than b, including b | [latex]\left(-\infty ,b\right][/latex] |
| [latex]{a}\le{x}\lt{ b}[/latex] | All real numbers between a and b, including a | [latex]\left[a,b\right)[/latex] |
| [latex]{a}\lt{x}\le{ b}[/latex] | All real numbers between a and b, including b | [latex]\left(a,b\right][/latex] |
| [latex]{a}\le{x}\le{ b}[/latex] | All real numbers between a and b, including a and b | [latex]\left[a,b\right][/latex] |
| [latex]{x}\lt{a}\text{ or }{x}\gt{ b}[/latex] | All real numbers less than a or greater than b | [latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex] |
| All real numbers | All real numbers | [latex]\left(-\infty ,\infty \right)[/latex] |
The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:
| Start With | Multiply By | Final Inequality |
| [latex]a>b[/latex] | [latex]c[/latex] | [latex]ac>bc[/latex] |
| [latex]5>3[/latex] | [latex]3[/latex] | [latex]15>9[/latex] |
| [latex]a>b[/latex] | [latex]-c[/latex] | [latex]-ac<-bc[/latex] |
| [latex]5>3[/latex] | [latex]-3[/latex] | [latex]-15<-9[/latex] |
The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:
| Start With | Divide By | Final Inequality |
| [latex]a>b[/latex] | [latex]c[/latex] | [latex] \displaystyle \frac{a}{c}>\frac{b}{c}[/latex] |
| [latex]4>2[/latex] | [latex]2[/latex] | [latex] \displaystyle \frac{4}{2}>\frac{2}{2}[/latex] |
| [latex]a>b[/latex] | [latex]-c[/latex] | [latex] \displaystyle -\frac{a}{c}<-\frac{b}{c}[/latex] |
| [latex]4>2[/latex] | [latex]-2[/latex] | [latex] \displaystyle -\frac{4}{2}<-\frac{2}{2}[/latex] |
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