1.3: Inequalities and intervals
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There is an order relation on the set of real numbers:
4<9 reads as 4 is less than 9,
−3≤2 reads as −3 is less than or equal to 2,
76>1 reads as 76 is greater than 1,
2≥−3 reads as 2 is greater than or equal to −3.
We have 2<3, but −2>−3, which can be seen on the number line above.
We have 5≤5 and 5≥5. However the same is not true when using the symbol <. We write this as 5≮5.
The set of all real numbers x greater than or equal to some number a and/or less than or equal to some number b is denoted in different ways by the following chart:
Inequality notation | Number line | Interval notation |
---|---|---|
a≤x≤b | ![]() |
[a,b] |
a<x<b | ![]() |
(a,b) |
a≤x<b | ![]() |
[a,b) |
a<x≤b | ![]() |
(a,b] |
a≤x | ![]() |
[a,∞) |
a<x | ![]() |
(a,∞) |
x≤b | ![]() |
(−∞,b] |
x<b | ![]() |
(−∞,b) |
Formally, we define the interval [a,b] to be the set of all real numbers x such that a≤x≤b:
[a,b]={x∣a≤x≤b}
There are similar definitions for the other intervals shown in the above table.
Be sure to write the smaller number a<b first when writing an interval [a,b]. For example, the interval [5,3]={x∣5≤x≤3} would be the empty set!
Graph the the inequality π<x≤5 on the number line and write it in interval notation.
Solution
On the number line:
Interval notation: (π,5]
Write the following interval as an inequality and in interval notation:
Solution
Inequality notation: −3≤x
Interval notation: [−3,∞)
Write the following interval as an inequality and in interval notation:
Solution
Inequality notation: & x<2
Interval notation: & (−∞,2)
In some texts round and square brackets are also used on the number line to depict an interval. For example the following displays the interval [2,5).