Consider the following inequality:
We can represent this inequality using a number line as follows:
How do we represent this inequality using interval notation?
Note the following:
The left endpoint is not included. Therefore, we use a parenthesis for this endpoint.
The interval continues forever in the positive direction. To denote this, we must introduce the infinity symbol Infinity is not a number but a concept that means larger than any number.
We use a right parenthesis in conjunction with the symbol.
Therefore, we can express the inequality in interval notation as follows:
Watch Out! Whenever appears in an interval, it is always associated with a right parenthesis. Infinity is never associated with a bracket, even if the other endpoint is.
Express the inequality in interval notation.
The inequality corresponds to all numbers greater than or equal to A number line for this inequality is shown below:
From the segment on the number line, we have the following:
The left endpoint is included in the interval.
There is no right endpoint - it continues forever. So, we write the interval with infinity as the right endpoint, not included.
Therefore, using interval notation, we represent as
The inequality $x > 3$ can be expressed in interval notation as
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a
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$x\in(-3, \infty)$ |
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b
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$x\in[-3, \infty)$ |
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c
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$x\in(3, \infty)$ |
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d
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$x\in[3, \infty)$ |
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e
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$x\in(-\infty, 3)$ |
The inequality $t \ge -1$ can be expressed in interval notation as
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a
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$t\in(-1, \infty)$ |
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b
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$t\in(-\infty, -1)$ |
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c
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$t\in(1, \infty)$ |
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d
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$t\in[1, \infty)$ |
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e
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$t\in[-1, \infty)$ |
What inequality corresponds the interval
The statement means that lies between and where
the left endpoint is included, and
the right endpoint is not included.
Therefore, the inequality represented by this interval is
However, since all numbers are smaller than infinity, it suffices to write
or, equivalently,
The interval $x\in(10, \infty)$ corresponds to the inequality
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a
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$x > -10$ |
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b
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$x \ge 10$ |
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c
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$x > 10$ |
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d
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$x < 10$ |
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e
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$x \le 10$ |
The interval $x\in\left[-5, \infty\right)$ corresponds to the inequality $x$
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a
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b
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c
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d
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e
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The interval $x\in\left[-3, \infty\right)$ corresponds to the inequality
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a
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$x \le -3$ |
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b
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$x \ge 3$ |
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c
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$x \ge -3$ |
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d
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$x < -3$ |
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e
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$x > -3$ |
Now, consider the following inequality:
We can represent this inequality using a number line as follows:
To represent this inequality using interval notation, we note the following:
The right endpoint is not included. Therefore, we use a parenthesis for this endpoint.
The interval continues forever in the negative direction. To denote this, we use the negative infinity symbol We use a left parenthesis in conjunction with the negative infinity symbol.
Therefore, we can express the inequality in interval notation as follows:
Express the inequality in interval notation.
The inequality corresponds to all numbers less than or equal to A number line for this interval is shown below:
From the segment on the number line, we have the following:
There is no left endpoint - it continues forever. So, we write the interval with negative infinity as the left endpoint, not included.
The right endpoint is included in the interval.
Therefore, using interval notation, we represent as
The inequality $x < -7$ can be expressed in interval notation as
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a
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$x\in(-\infty, -7]$ |
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b
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$x\in[-7, \infty)$ |
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c
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$x\in(-7, \infty)$ |
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d
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$x\in(-\infty, -7)$ |
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e
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$x\in[7, \infty)$ |
The inequality $y \le 0$ can be expressed in interval notation as
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a
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$y\in(-\infty, 0]$ |
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b
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$y\in(-\infty, 0)$ |
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c
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$y\in(-\infty, 1]$ |
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d
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$y\in[0, \infty)$ |
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e
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$y\in[-\infty, 1]$ |
What inequality corresponds to the interval
The statement means that lies between and where
the left endpoint is not included, and
the right endpoint is not included.
Therefore, the inequality represented by this interval is
However, since all numbers are greater than negative infinity, it suffices to write
The interval notation $x\in\left(-\infty, -1\right)$ corresponds to the inequality $x$
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a
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b
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c
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d
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e
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The interval notation $y\in\left(-\infty, -\dfrac{7}{4}\right)$ corresponds to the inequality
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a
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$y > -\dfrac{7}{4}$ |
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b
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$y \leq -\dfrac{7}{4}$ |
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c
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$y < -\dfrac{7}{4}$ |
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d
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$y < -\dfrac{3}{4}$ |
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e
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$y \geq -\dfrac{7}{4}$ |
The interval notation $y\in\left(-\infty, -4\right]$ corresponds to the inequality $y$
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a
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b
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c
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d
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e
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