Question

Express the interval in set-builder notation and graph the interval on a number line. (-2, 3]

Answer 1

The interval
`\((-2, 3)\)` in set-builder notation is
`\(\{ x \mid -2 < x \leq 3 \}\),`and on the number line, it includes all real numbers greater than -2 and less than or equal to 3, with a hollow circle at -2 and a filled circle at 3.

The given interval
`\((-2, 3)\)` can be expressed in set-builder notation as follows:

`\[ \{ x \mid -2 < x \leq 3 \} \]`

This notation indicates that the set consists of all real numbers
`\(x\)` such that \(x\) is greater than -2 and less than or equal to 3.

To graph this interval on a number line, we represent the numbers between -2 and 3, including -2 (because of the open parenthesis), and excluding 3 (because of the closed bracket). A hollow circle is used for -2 to denote that it is not included, and a filled circle is used for 3 to indicate that it is included. A line connects these two points, covering all the values within the interval.

On the number line, it would look like this:

```

-----o------------------------o---------

-2 3

```

The interval includes all real numbers greater than -2 and less than or equal to 3. The open circle at -2 signifies that -2 is not included, and the closed circle at 3 indicates that 3 is included in the interval.