Suppose we want to represent the following number line graph using interval notation:
This number line consists of the interval as well as the interval
To express the combination of these two intervals, we create a union of intervals using the union sign as follows:
Express the compound inequality or using interval notation.
We can represent the compound inequality on a number line, as follows:
This number line consists of the interval as well as the interval
So, we can express the compound inequality as the union of these two intervals:
Express the compound inequality $x < -2$ or $x \ge 3$ using interval notation.
|
a
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$(-\infty, 2] \cup (3, \infty)$ |
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b
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$(-\infty, -2) \cup [3, \infty)$ |
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c
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$(-\infty, -2) \cup (3, \infty)$ |
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d
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$(-\infty, 2) \cup [3, \infty)$ |
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e
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$(-\infty, -2] \cup [3, \infty)$ |
Express the compound inequality $x \leq 0$ or $x > 2$ using interval notation.
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a
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$(-\infty, 0] \cup [2, \infty)$ |
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b
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$(0,2)$ |
|
c
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$(-\infty, 0) \cup (2, \infty)$ |
|
d
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$(-\infty, 0) \cup [2, \infty)$ |
|
e
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$(-\infty, 0] \cup (2, \infty)$ |
Express as an inequality.
First, we graph the two intervals on a number line:
The interval can be expressed as the inequality
Likewise, the interval can be expressed as the inequality
So, the union of these intervals represents the compound inequality
Express $(-\infty, 1] \cup [3, \infty)$ as an inequality.
|
a
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$x > 1\, $ or $\, x < 3$ |
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b
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$x < 1\, $ or $\, x \geq 3$ |
|
c
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$x \geq 1\, $ or $\, x \geq 3$ |
|
d
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$x \leq 1\, $ or $\, x \geq 3$ |
|
e
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$x < -1\, $ or $\, x \geq -3$ |
Express $(-\infty, -7) \cup (0, \infty)$ as an inequality.
|
a
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$-7 < x \le 0$ |
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b
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$x < -7$ or $x \ge 0$ |
|
c
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$x \le -7$ or $x < 0$ |
|
d
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$x < -7$ or $x > 0$ |
|
e
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$-7 < x < 0$ |
Sometimes, the union of two intervals can simplify into a single interval. For example, consider the union
If we plot these two intervals on number lines, we get the following:
The union of the two intervals consists of all the points which are contained within at least one of the intervals. So, to graph the union of the two intervals, we take the graphs of the individual intervals and lay them on top of each other.
The resulting graph is as follows:
This graph represents the interval So, the union simplifies to the interval
What is the union of the intervals shown on the number lines below?
We have the following number lines:
The union of the two intervals consists of all the points which are contained within at least one of the intervals. So, to graph the union of the two intervals, we take the graphs of the individual intervals and lay them on top of each other.
The resulting graph is as follows:
The graph represents the interval
What is the union of the intervals shown on the number lines above?
|
a
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$x \geq -1$ |
|
b
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$x > -1$ |
|
c
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$x \geq 1$ |
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d
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$- 1 \leq x < 1$ |
|
e
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$x>1$ |
What is the union of the intervals shown on the number lines above?
|
a
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$(-2,2)$ |
|
b
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$(-3 , 5]$ |
|
c
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$[-5,-3)$ |
|
d
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$[-5,2]$ |
|
e
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$(-3,-2)$ |
Simplify the union of intervals
If we plot these two intervals on number lines, we get the following:
The union of the two intervals consists of all the points which are contained within at least one of the intervals. So, to graph the union of the two intervals, we take the graphs of the individual intervals and lay them on top of each other.
The resulting graph is as follows:
This graph represents the interval So, the union simplifies to the interval
Simplify the union of intervals $(-\infty, 5] \cup (3, \infty).$
|
a
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$(-\infty,\infty)$ |
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b
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$[5, \infty)$ |
|
c
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$(-\infty, 5]$ |
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d
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$[3,5)$ |
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e
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$(5, \infty)$ |
Simplify the union of intervals $(-2, 3] \cup (1, \infty).$
|
a
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$(1,3]$ |
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b
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$(-2,\infty)$ |
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c
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$(-2,1)$ |
|
d
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$[-2,\infty)$ |
|
e
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$[3,\infty)$ |
