Using Interval Notation
Indicating the solution to an inequality such as [latex]x\ge 4[/latex] can be achieved in several ways. We can use a number line as shown in Figure 2. The blue ray begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
Set Indicated | Set-Builder Notation | Interval Notation |
---|---|---|
All real numbers between a and b, but not including a or b | [latex]\{x|a<x<b\}[/latex] | [latex]\left(a,b\right)[/latex] |
All real numbers greater than a, but not including a | [latex]\{x|x>a\}[/latex] | [latex]\left(a,\infty \right)[/latex] |
All real numbers less than b, but not including b | [latex]\{x|x<b\}[/latex] | [latex]\left(-\infty ,b\right)[/latex] |
All real numbers greater than a, including a | [latex]\{x|x\ge a\}[/latex] | [latex]\left[a,\infty \right)[/latex] |
All real numbers less than b, including b | [latex]\{x|x\le b\}[/latex] | [latex]\left(-\infty ,b\right][/latex] |
All real numbers between a and b, including a | [latex]\{x|a\le x<b\}[/latex] | [latex]\left[a,b\right)[/latex] |
All real numbers between a and b, including b | [latex]\{x|a<x\le b\}[/latex] | [latex]\left(a,b\right][/latex] |
All real numbers between a and b, including a and b | [latex]\{x|a\le x\le b\}[/latex] | [latex]\left[a,b\right][/latex] |
All real numbers less than a or greater than b | [latex]\{x|x<a\text{ and }x>b\}[/latex] | [latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex] |
All real numbers | [latex]\{x|x\text{ is all real numbers}\}[/latex] | [latex]\left(-\infty ,\infty \right)[/latex] |
Example 1: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a
Use interval notation to indicate all real numbers greater than or equal to [latex]-2[/latex].Solution
Use a bracket on the left of [latex]-2[/latex] and parentheses after infinity: [latex]\left[-2,\infty \right)[/latex]. The bracket indicates that [latex]-2[/latex] is included in the set with all real numbers greater than [latex]-2[/latex] to infinity.Try It 1
Use interval notation to indicate all real numbers between and including [latex]-3[/latex] and [latex]5[/latex]. SolutionExample 2: Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b
Write the interval expressing all real numbers less than or equal to [latex]-1[/latex] or greater than or equal to [latex]1[/latex].Solution
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\infty [/latex] and ends at [latex]-1[/latex], which is written as [latex]\left(-\infty ,-1\right][/latex]. The second interval must show all real numbers greater than or equal to [latex]1[/latex], which is written as [latex]\left[1,\infty \right)[/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\cup [/latex], between the two intervals.[latex]\left(-\infty ,-1\right]\cup \left[1,\infty \right)[/latex]