February 25, 2021

Chapters

## Interval Definition

Intervals are a very important idea in mathematics. The **basic definition** of an interval is the numbers inside of a range. A range is usually made up of two numbers. Take a look at the image below, which gives some real life examples of intervals.

Interval | Type | |

A | From 14:00 to 15:00 | Time |

B | From a grade of F of 0 score to an A of 90 points | Test Scores |

C | From 18 years old to 64 | Age |

## Interval properties

As you saw in the previous section, intervals can be almost anything: time, age, test scores and more. However, there are a few things you should keep in mind when creating your own interval. The most important property of an interval is that it includes all **real numbers.**

**example**of real numbers. These are explained in the table below.

Type | Explanation | |

A | Whole number | No decimals, positive |

B | Rational number | Positive numbers with decimals or fractions |

C | Rational number | Can also be negative |

D | Irrational number | Can’t be written as a decimal |

## Including Ends

The most important thing to remember is the notation of intervals on the line itself and when written in numbers. On a number line, we usually depict intervals as a straight line with **two circles** at the end. When the circle is **filled** in, this means that we want to include that number in our interval.

**start**at 20 and

**end**with 40, rather than starting at 21 and ending at 39.

Because drawing a number line every time we want to express this is inconvenient, we can also simply **write:**

**same**thing as the filled in circle: it includes 20 and 40.

## Excluding Ends

When we want to exclude the ends, we can simply leave the circles **empty.** Let’s take the same example above, only this time, say we only want to include 40.

**21**and finish with

**40.**When we want to exclude a number from our interval, we simply use parenthesis.

## Inequalities

Inequalities deal with two ideas: **greater** and **less** than. The table below shows the definitions of both.

A | Less Than | < | When a number, x, is to the left of another number, y, on a number line | 2 < 10 |

B | Greater Than | > | When a number, x, is to the right of another number, y, on a number line | 10 > 2 |

We can also combine inequalities with an equals sign. Check out the possible combinations below.

A | Less than OR equal to | When a number is to the left of or equal to another on a number line | 2 10, 10 10 | |

B | Greater than OR equal to | When a number is to the right of or equal to another on a number line | 10 2, 10 10 |

## Infinity

Infinity is something that has **no end.** Infinity is endless and is a very helpful concept in maths. When we talk about infinity in inequalities, we always exclude infinity. This is because we **never reach** infinity - it is endless after all!

**parenthesis.**Let's take [3, +) as another example. When we draw this on a number line, instead of using an empty circle, we simply use an arrow.

## Notation

Now that we have learned about intervals and inequalities, we can **combine** both to understand statements of inequalities. Say, for example, that you have a birthday coming up. You know you will get between 100 to 200 pounds from your family and friends. This can be written as:

**between**100 and 200 because it can be anywhere: 101, 115, 199, and so on. Take a look at the image below and the table that describes the image.

A | a < x < b | (a,b) | Exclude a and b |

B | a > x > b | (a,b) | Exclude a and b |

C | a x b | [a,b] | Include a and b |

D | a x b | [a,b] | Include a and b |

## Union

A union can be thought of as combining **several expressions** of inequality. The easiest way to understand this is to use infinity. For example, Antoine and Marie are playing a video game where you have positive and negative points. The table describes their possible results.

Antoine | (- , 10] | We know Antoine scored 10 points or less |

Marie | (7, + ) | We know Marie scored above 7 points |

If we wanted to find the region where they could have tied, we can make a union of these two intervals. A **union** is written as:

**number line.**

**intersection.**

## Intersection

An intersection is exactly as it sounds: it is where two things **meet.** In this case, it is where our two intervals meet. Lets continue from the example above.

Here are the possible **scenarios** of this game.

Antoine (A) wins | A = 10 | If Antoine scores 10 points |

Marie (M) wins | A > 10 | If Marie scores more than 10 points |

Antoine and Marie tie | 7 < A 10 7 < M 10 | If both Antoine and Marie score above 7 points and 10 points or less |

We can break the table down using the number line.

A | A < 7 | Antoine loses, as his interval is the only one going below 7 |

B | 7 < A 10 7 < M 10 | Both Antoine and Marie win, as their intervals both intersect each other between these two points |

C | M 10 | Marie wins, as her interval is the only one at 10 or above |

As you can see, the region where they tie is the intersection because it is the interval where both Antoine and Marie’s intervals meet each other. Taking away the **A** and **C** parts of the number line, we are left with:

Which gives us the following:

Notation | Intersection | Description |

7 < x 10 | (7,10] | Exclude 7 and include 10 |