Inequalities

Lesson

An *interval *on the real number line is the set of numbers between two endpoints. One or both endpoints, or neither, can belong to interval and there are notations for each possibility.

Intervals that include their endpoints are called *closed* intervals. To specify a closed interval including all numbers between $-1$−1 and $2\pi$2π, for example, we write $\left[-1,2\pi\right]$[−1,2π]. The square brackets indicate that the numbers $-1$−1 and $2\pi$2π are considered to belong to the interval.

Intervals that do not include their endpoints are called *open* intervals. The notation for these uses rounded brackets. For example, we write $\left(0,100\right)$(0,100) to mean the set of numbers between $0$0 and $100$100 but not including either $0$0 or $100$100.

Open intervals do not have a maximum or a minimum element. This is because however close to an endpoint a number in the interval is, there is always another number that is closer to the endpoint. Instead, it may be appropriate to refer to the endpoints respectively as the least upper bound and greatest lower bound of the interval.

It is also possible for an interval to be closed at one end but open at the other. For example, $[0,\sqrt{2})$[0,√2) or $(-9,0]$(−9,0].

An interval with no upper bound is indicated with the $\left[a,\infty\right)$[`a`,∞) sign. Such intervals are said to be open on the right. Similarly, an interval with no lower bound is open on the left and is notated with the sign $\left(-\infty,b\right]$(−∞,`b`].

Thus, for example, we can indicate the whole real number line with the notation $\left(-\infty,\infty\right)$(−∞,∞).

To make a statement that a number $x$`x` is in a certain interval, we can write $x\in(a,b)$`x`∈(`a`,`b`). We understand this to mean that $x$`x` is greater than $a$`a` and $x$`x` is less than $b$`b`.

That is, using the usual notations for 'greater than' and 'less than', the statement says $x>a$`x`>`a` and $x`x`<`b`. We could also write $a`a`<`x` and $x`x`<`b` and then combine these two statements into one as $a`a`<`x`<`b`, meaning that $x$`x` is strictly between $a$`a` and $b$`b`.

These forms of notation using $<$< and $>$> are called *inequalities*.

The notation $a`a`<`x`<`b` is equivalent to the notation $x\in(a,b)$`x`∈(`a`,`b`).

We also use $a\le x\le b$`a`≤`x`≤`b` to indicate that $x$`x` is in the closed interval from $a$`a` to $b$`b`. That is, $x\in\left[a,b\right]$`x`∈[`a`,`b`].

Sometimes, an interval might be illustrated with a diagram like the following.

The diagram represents the interval $[-3,1)$[−3,1). A solid endpoint indicates that the point is included and a hollow endpoint indicates that the point is not included.

To specify that a number $x$`x` belongs to this interval, we have the option of writing $x\in[-3,1)$`x`∈[−3,1) or we can use the pair of inequalities $-3\le x$−3≤`x` and $x<1$`x`<1, which can be combined as $-3\le x<1$−3≤`x`<1.

The following diagram illustrates an interval that has no upper bound. It would be notated either $x\ge-1$`x`≥−1 or $x\in[-1,\infty)$`x`∈[−1,∞).

Sometimes we wish to say that a number $x$`x` can be in one of two (or more) disjoint intervals. It may be that $x<-1$`x`<−1 or $x>1$`x`>1. This can also be notated using the sign for set union: $x\in(-\infty,-1)\cup(1,\infty)$`x`∈(−∞,−1)∪(1,∞).

Consider the pictured inequality.

The endpoints of this interval, from left to right, are $\editable{}$ and $\editable{}$.

Which of the following is the correct notation for the pictured interval?

$[$[$-1$−1,$2$2$]$]

A$($($-1$−1,$2$2$]$]

B$[$[$-1$−1,$2$2$)$)

C$($($-1$−1,$2$2$)$)

D

Consider the pictured inequality.

The endpoints of this interval are $\editable{}$ and $\infty$∞.

Which of the following is the correct notation for the pictured interval?

$[$[$-3$−3,$\infty$∞$]$]

A$[$[$-3$−3,$\infty$∞$)$)

B$($($-3$−3,$\infty$∞$)$)

C$($($-3$−3,$\infty$∞$]$]

D

Consider the inequality $x\le6$`x`≤6.

The endpoints of this interval are $-\infty$−∞ and $\editable{}$.

Which of the following is the correct notation for the given inequaliaty?

$($($-\infty$−∞,$6$6$]$]

A$($($-\infty$−∞,$6$6$)$)

B$[$[$-\infty$−∞,$6$6$]$]

C$[$[$-\infty$−∞,$6$6$)$)

D