- Last updated

- Save as PDF

- Page ID
- 83128

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vectorC}[1]{\textbf{#1}}\)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

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: | |

Either \(≤\) or \(≥\) | Left 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.

- \(x ≥ 4\) or \(x ≤ 0\)
- \(x ≤ – 2\pi\) or \(x > \pi\)
- \(−1 > x\) or \(2 ≤ x\)
- \(x > 3\pi\) or \(x < – \pi\)