# 4.2: Interval Notation

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Inequalities slice and dice the real number line into segments of interest or intervals. An interval is a continuous, uninterrupted subset of real numbers. How can we notate intervals with simplicity? The table below introduces Interval Notation.

Inequality Associated Circle Associated Endpoint Closures

Either $$<$$ or $$>$$

Left Parenthesis: ( or Right Parenthesis: )

Either $$≤$$ or $$≥$$

Left Square bracket: [ or Right Square Bracket: ]

Inequalities have $$4$$ possible interval closures:

 $$(A,B)$$ $$(A,B]$$ $$[A,B)$$ $$[A,B]$$

The least number in the interval, $$A$$, is always stated first. A comma is placed. The largest number in the interval, $$B$$, is stated after the comma. The appropriate closure is considered for each value $$A$$ and $$B$$.

## Four Examples of Interval Notation

 $$−2 < x < 3$$ $$−2 < x ≤ 3$$ $$– 2 ≤ x < 3$$ $$– 2 ≤ x ≤ 3$$ $$(−2, 3)$$ $$(−2, 3]$$ $$[−2, 3)$$ $$[−2, 3]$$

## The Infinities

There are two infinities: positive and negative. Each define a direction on the number line:

Infinity is not a real number. It indicates a direction. Therefore, when using interval notation, always enclose $$∞$$ and $$−∞$$ with parenthesis. We never enclose infinities with square bracket.

The table below shows four examples of interval notation that require the use of infinity.

 $$x < 2$$ $$x ≤ 2$$ $$x > 2$$ $$x ≥ 2$$ $$(−∞, 2)$$ $$(−∞, 2]$$ $$(2, ∞)$$ $$[2, ∞)$$

## Combinations of Intervals

If two or more intervals are interrupted with a gap in the number line, set notation is used to stitch the intervals together, symbolically. The symbol we use to combine intervals is the union symbol: $$∪$$. The table below shows four examples:

Interval Notation Graph
$$(−∞, −2) ∪ [1, ∞)$$
$$(−∞, −1) ∪ (−1, ∞)$$
$$\left(−\dfrac{3 \pi}{2} , −\dfrac{\pi}{2} \right) ∪ \left( \dfrac{\pi}{2}, \dfrac{3 \pi}{2} \right)$$
$$\left[−2 \pi, − \dfrac{\pi}{2} \right) ∪ \left[ \dfrac{3 \pi}{2} , ∞ \right)$$

## Compound Inequalities

Intervals that have gaps, like the ones shown above, translate to compound inequalities. Real solutions belong in one interval or another. The word “or” plays a key role when translating. For example: the interval $$(−∞, −2) ∪ [1, ∞)$$ translates to its associated compound inequality:

$$x < -2$$ or $$x ≥ 1$$

The word “and” cannot be used between the inequalities because a number cannot belong to both intervals at once. For example, $$x = 5$$ is a solution because $$5$$ belongs in the interval $$x ≥ 1$$, but $$5$$ does not belong in the interval $$x < −2$$. Nevertheless, because of the word “or,” $$x = 5$$ is a solution to the interval $$(−∞, −2) ∪ [1, ∞)$$.

## Try It! (Exercises)

For exercises #1-6, state the inequality and the interval notation associated with the graph.

Graph Inequality Interval Notation

For exercises #7-10, state the interval notation and sketch the graph associated with the inequality.

Graph Inequality Interval Notation
$$−3 ≤ x ≤ 1$$
$$x < 4$$
$$x ≥ −2$$
$$0 ≤ x < 3$$

For exercises #11-17, sketch the graph associated with the given interval notation.

Graph Interval Notation
$$(−∞, 4)$$
$$(−∞, −3) ∪ [0, ∞)$$
$$[−1, 1) ∪ [2, ∞)$$
$$(−∞, −5] ∪ (−1, 5)$$
$$\left[−\dfrac{\pi}{2} , \dfrac{\pi}{2} \right]$$
$$(−∞, −\pi] ∪ [\pi, ∞)$$
$$\left(−\dfrac{3\pi}{2} , −\dfrac{\pi}{2} \right) ∪ \left(-\dfrac{\pi}{2} , 0\right)$$

For #18-21

a. Sketch a graph of the compound inequality.

b. State the interval using interval notation.

1. $$x ≥ 4$$ or $$x ≤ 0$$
2. $$x ≤ – 2\pi$$ or $$x > \pi$$
3. $$−1 > x$$ or $$2 ≤ x$$
4. $$x > 3\pi$$ or $$x < – \pi$$

This page titled 4.2: Interval Notation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.