Consider the region below, which represents the inequality
We can represent this inequality using interval notation as follows:
The parentheses tell us that the endpoints and are not included in this interval.
When an interval does not include the endpoints, we say that it is open.
More formally, we can also write
The symbol means "lies in" or "belongs to." So, the above reads, " belongs to the (open) interval "
Now consider the region below, which represents the inequality
We express this inequality using interval notation as follows:
The brackets tell us that the endpoints and are included in this interval.
When an interval includes the endpoints, we say that it is closed.
More formally, we can also write
The above reads, " belongs to the (closed) interval "
Which inequality corresponds to the interval
Every interval can be expressed as an inequality. We use the following notations:
Parentheses to indicate that the endpoints are not included.
Brackets to indicate that the endpoints are included.
Our interval contains only parentheses. So, the endpoints are not included.
Therefore, our interval can be expressed as the following inequality:
The inequality $3 < x < 7$ can be expressed in interval notation as
|
a
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$(-\infty, 7)$ |
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b
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$(3, 7)$ |
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c
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$[3, 7)$ |
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d
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$[3, 7]$ |
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e
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$(3, 7]$ |
Which inequality corresponds to the interval $x \in (3, 4)?$
|
a
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$x< 3$ or $x>4$ |
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b
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$3 \leq x < 4$ |
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c
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$3 < x \leq 4$ |
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d
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$3 < x < 4$ |
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e
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$3 \leq x \leq 4$ |
Express the inequality in interval notation.
Every inequality can be expressed as an interval. We use the following notations:
Parentheses if the endpoints are not included.
Brackets if the endpoints are included.
Our inequality contains only "or equal to" symbols. So, the endpoints and are included.
Therefore, we can express our inequality in interval notation as follows:
More formally, we can also write
The inequality $-10 \le x \le 3$ can be expressed in interval notation as
|
a
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$[-10,3]$ |
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b
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$[-10,3)$ |
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c
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$(-10,3]$ |
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d
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$(-\infty,3]$ |
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e
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$(-10,3)$ |
Which inequality corresponds to the interval $x \in [-2, 1]?$
|
a
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$-2 < x \leq 1$ |
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b
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$-2 < x < 1$ |
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c
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$-2 \leq x < 1$ |
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d
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$-2 \leq x \leq 1$ |
|
e
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$x < -2$ or $x > 1$ |
Some intervals may include one parenthesis and one bracket.
For example, consider the following interval:
To express this interval as an inequality, we note the following:
The left bracket tells us that left endpoint is included in the interval. Therefore, we use for this endpoint.
The right parenthesis tells us that the right endpoint is not included in the interval. Therefore, we use for this endpoint.
Therefore, the inequality that corresponds to the given interval is
The interval can be graphed as follows:
Intervals with one bracket and one parenthesis are called half-open (or half-closed).
Write the interval as an inequality.
Every interval can be expressed as an inequality. We use the following notations:
Parentheses to indicate that the endpoints are not included.
Brackets to indicate that the endpoints are included.
For the given interval
we note the following:
The left endpoint is not included in the interval. Therefore, we use
The right endpoint is included in the interval. Therefore, we use
Therefore, the inequality that corresponds to the given interval is
The inequality $9 < x \le 13$ can be expressed in interval notation as
|
a
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$x\in[9, 13)$ |
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b
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$x\in(9, 13)$ |
|
c
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$x\in(9, 13]$ |
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d
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$x\in(13, 9]$ |
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e
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$x\in[9, 13]$ |
The interval $y\in\big[-5,\, 2\sqrt{2}\,\big)$ corresponds to the inequality
|
a
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$-5 \leq y < 2\sqrt{2}$ |
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b
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$-5 \leq y \leq 2\sqrt{2}$ |
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c
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$-5 < y \leq 2\sqrt{2}$ |
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d
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$5 \leq y < 2\sqrt{2}$ |
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e
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$-5 < y < 2\sqrt{2}$ |
What interval corresponds to the segment shown below?
Every inequality can be expressed as an interval. We use the following notations:
Parentheses if the endpoints are not included.
Brackets if the endpoints are included.
The given interval can be represented by the following inequality:
We can express our inequality in interval notation as follows:
Which of the following intervals corresponds to the segment shown above?
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a
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$x\in[1,4)$ |
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b
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$x\in[1,4]$ |
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c
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$x\in(-1,4)$ |
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d
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$x\in(1,4)$ |
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e
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$x\in(1,4]$ |
Which of the following intervals corresponds to the segment shown above?
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a
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$x\in[-1,2]$ |
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b
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$x\in(-1,2]$ |
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c
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$x\in[-1,2)$ |
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d
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$x\in(-\infty,2]$ |
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e
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$x\in(-1,2)$ |