To solve the problem of graphing the set [tex]\( \{x \mid 2 < x \leq 5\} \)[/tex] on the number line and writing the set using interval notation, follow these detailed steps:
1. Understanding the Set Notation:
- The set [tex]\( \{x \mid 2 < x \leq 5\} \)[/tex] includes all real numbers [tex]\( x \)[/tex] that are strictly greater than 2 and up to and including 5.
2. Graphing on the Number Line:
- Draw a horizontal number line.
- Identify the points 2 and 5 on the number line.
- Because [tex]\( x \)[/tex] is strictly greater than 2, place an open circle at 2. This indicates that 2 is not included in the set.
- Because [tex]\( x \)[/tex] is less than or equal to 5, place a closed circle at 5. This indicates that 5 is included in the set.
- Shade the region on the number line between the open circle at 2 and the closed circle at 5 to represent all the numbers in between.
- Your number line should look like this:
[tex]\[
\begin{array}{c}
\text{< - - - -} \circ \text{==================}\bullet \text{- - - - >}\\
\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}2\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}\phantom{0}5\phantom{0}\phantom{0}\phantom{0}\phantom{0}
\end{array}
\][/tex]
3. Writing the Interval Notation:
- Interval notation expresses the set in terms of its lower and upper bounds.
- For the set [tex]\(2 < x \leq 5\)[/tex], the lower bound is 2 and the upper bound is 5.
- Use a parenthesis [tex]\( ( \)[/tex] to indicate that 2 is not included.
- Use a bracket [tex]\( ] \)[/tex] to indicate that 5 is included.
- Therefore, the interval notation for the set [tex]\( \{x \mid 2 < x \leq 5\} \)[/tex] is [tex]\( (2, 5] \)[/tex].
In conclusion, the interval notation for the set [tex]\( \{x \mid 2 < x \leq 5\} \)[/tex] is [tex]\( (2, 5] \)[/tex]. The number line representation with an open circle at 2, a shaded region up to 5, and a closed circle at 5 is also complete.
