1.8: Intervals and Interval Notation
Media
| Title | Level of Approach | Type | Length |
|---|---|---|---|
| Interval Notation | Elementary Algebra | Lecture | 15:05 |
| A Review of Interval Notation | Intermediate Algebra and Beyond | Review | 13:30 |
Using Interval Notation
Indicating the solution to an inequality such as \(x \geq 4\) can be achieved in several ways.
- We can use a number line as shown in Figure \(\PageIndex{1}\). The blue ray begins at \(x = 4\) and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to \(4\).
- We can use set-builder notation : \(\{x \mid x \geq 4\}\), which translates to "all real numbers \(x\) such that \(x\) is greater than or equal to \(4\)." Notice that braces are used to indicate a set.
- The third method is interval notation , in which solution sets are indicated with parentheses or brackets. The solutions to \(x \geq 4\) are represented as \([4,\infty)\).
Figure \(\PageIndex{1}\)
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be "equaled." A few examples of an interval, or a set of numbers in which a solution falls, are \([−2,6)\), or all numbers between \(−2\) and \(6\), including \(−2\), but not including \(6\); \((−1,0)\), all real numbers between, but not including \(−1\) and \(0\); and \((−\infty,1]\), all real numbers less than and including \(1\). Table \(\PageIndex{1}\) outlines the possibilities.
| Set Indicated | Set-Builder Notation | Interval Notation |
|---|---|---|
| All real numbers between \(a\) and \(b\), but not including \(a\) or \(b\) | \(\{x \mid a<x<b\}\) | \((a,b)\) |
| All real numbers greater than \(a\), but not including \(a\) | \(\{x \mid x>a\}\) | \((a,\infty)\) |
| All real numbers less than \(b\), but not including \(b\) | \(\{x \mid x<b\}\) | \((−\infty,b)\) |
| All real numbers greater than \(a\), including \(a\) | \(\{x \mid x \geq a\}\) | \([a,\infty)\) |
| All real numbers less than \(b\), including \(b\) | \(\{x \mid x \leq b\}\) | \((−\infty,b]\) |
| All real numbers between \(a\) and \(b\), including \(a\) | \(\{x \mid a \leq x<b\}\) | \([a,b)\) |
| All real numbers between \(a\) and \(b\), including \(b\) | \(\{x \mid a<x \leq b\}\) | \((a,b]\) |
| All real numbers between \(a\) and \(b\), including \(a\) and \(b\) | \(\{x \mid a \leq x \leq b\}\) | \([a,b]\) |
| All real numbers less than \(a\) or greater than \(b\) | \(\{x \mid x<a\space and\space x>b\}\) | \((−\infty,a)\cup(b,\infty)\) |
| All real numbers | \(\{x \mid x\space is\space all\space real\space numbers\}\) | \((−\infty,\infty)\) |
Use interval notation to indicate all real numbers greater than or equal to \(−2\).
- Solution
-
Use a bracket on the left of \(−2\) and parentheses after infinity: \([−2,\infty)\). The bracket indicates that \(−2\) is included in the set with all real numbers greater than \(−2\) to infinity.
Use interval notation to indicate all real numbers between and including \(−3\) and \(5\).
- Answer
-
\([−3,5]\)
Write the interval expressing all real numbers less than or equal to \(−1\) or greater than or equal to \(1\).
- 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 \(−\infty\) and ends at \(−1\), which is written as \((−\infty,−1]\).
The second interval must show all real numbers greater than or equal to \(1\), which is written as \([1,\infty)\). However, we want to combine these two sets. We accomplish this by inserting the union symbol, \cup , between the two intervals.
\[(−\infty,−1]\cup[1,\infty) \nonumber\]
Express all real numbers less than \(−2\) or greater than or equal to \(3\) in interval notation.
- Answer
-
\((−\infty,−2)\cup[3,\infty)\)