Represent the following sets on a number line: \( \begin{array}{llll}\text { (1) } & \{x:-14 \frac{1}{2} ; x \in \mathbb{R}\right\} & \text { (6) } & \{x: x \leq 7 ; x \in \mathbb{Z}\} \\ \text { (7) } & \{x: x<7 ; x \in \mathbb{N}\} & \text { (8) } & \{x: x \geq 0 ; x \in \mathbb{Z}\}\end{array} \)

Sure! Below are descriptions of how each set would be represented on a number line. Since I can't provide actual drawings, I'll describe the key features for each representation. --- ### **(1) \(\{x \mid -1 < x < 12 ;\, x \in \mathbb{Z}\}\)** - **Description**: This set includes all **integer** values of \(x\) that are greater than \(-1\) and less than \(12\). - **Elements**: \( \{0, 1, 2, \dots, 11\} \) - **Number Line Representation**: - **Points**: Place solid dots at each integer from **0** to **11**. - **Excluded Points**: Do **not** include \(-1\) and \(12\) since the inequalities are strict. - **Visualization**: ``` -----•-----•-----•-----•-----•-----•-----•-----•-----•-----•-----•-----• 0 1 2 3 4 5 6 7 8 9 10 11 ``` --- ### **(2) \(\{x \mid x \geq -4 ;\, x \in \mathbb{R}\}\)** - **Description**: This set includes all **real** numbers \(x\) that are greater than or equal to \(-4\). - **Number Line Representation**: - **Endpoint**: Place a **filled circle** at \(-4\) to indicate inclusion. - **Range**: Draw a solid line (or arrow) extending to the right from \(-4\) towards positive infinity. - **Visualization**: ``` <—–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– -4●——————————————————————————————> ``` --- ### **(3) \(\{x \mid -3 \leq x < 1 ;\, x \in \mathbb{R}\}\)** - **Description**: This set includes all **real** numbers \(x\) from \(-3\) up to but not including \(1\). - **Number Line Representation**: - **Start Point**: Place a **filled circle** at \(-3\) to indicate inclusion. - **End Point**: Place an **open circle** at \(1\) to indicate exclusion. - **Range**: Draw a solid line between \(-3\) and \(1\). - **Visualization**: ``` -3●——————————————o1 ``` --- ### **(4) \(\{x \mid x < 2 ;\, x \in \mathbb{R}\}\)** - **Description**: This set includes all **real** numbers \(x\) that are less than \(2\). - **Number Line Representation**: - **Endpoint**: Place an **open circle** at \(2\) to indicate exclusion. - **Range**: Draw a solid line (or arrow) extending to the left from \(2\) towards negative infinity. - **Visualization**: ``` <——————————————o2 ``` --- ### **(5) \(\left\{x \mid x > 4 \dfrac{1}{2} ;\, x \in \mathbb{R}\right\}\)** - **Description**: This set includes all **real** numbers \(x\) that are greater than \(4.5\). - **Number Line Representation**: - **Endpoint**: Place an **open circle** at \(4.5\) to indicate exclusion. - **Range**: Draw a solid line (or arrow) extending to the right from \(4.5\) towards positive infinity. - **Visualization**: ``` 4.5o————————————————————————> ``` --- ### **(6) \(\{x \mid x \leq 7 ;\, x \in \mathbb{Z}\}\)** - **Description**: This set includes all **integer** values \(x\) that are less than or equal to \(7\). - **Elements**: \( \{\dots, -2, -1, 0, 1, 2, \dots, 6, 7\} \) - **Number Line Representation**: - **Endpoint**: Place a **filled circle** at \(7\) to indicate inclusion. - **Range**: Draw solid dots at each integer to the left of \(7\). Use an arrow pointing to the left from a starting point (e.g., beyond \(-2\)) to indicate continuation towards negative infinity. - **Visualization**: ``` <————————•—•—•—•—•—•—•7● ``` --- ### **(7) \(\{x \mid x < 7 ;\, x \in \mathbb{N}\}\)** - **Description**: This set includes all **natural numbers** \(x\) that are less than \(7\). Assuming \(\mathbb{N} = \{1, 2, 3, \dots\}\). - **Elements**: \( \{1, 2, 3, 4, 5, 6\} \) - **Number Line Representation**: - **Points**: Place solid dots at each natural number from **1** to **6**. - **Excluded Point**: Do **not** include \(7\). - **Visualization**: ``` •1 •2 •3 •4 •5 •6 ``` --- ### **(8) \(\{x \mid x \geq 0 ;\, x \in \mathbb{Z}\}\)** - **Description**: This set includes all **integer** values \(x\) that are greater than or equal to \(0\). - **Elements**: \( \{0, 1, 2, 3, \dots\} \) - **Number Line Representation**: - **Endpoint**: Place a **filled circle** at \(0\) to indicate inclusion. - **Range**: Draw solid dots at each integer to the right of \(0\). Use an arrow pointing to the right from \(0\) to indicate continuation towards positive infinity. - **Visualization**: ``` 0●—•—•—•—•—•—•> ``` --- **Key Symbols Used:** - **• (Solid Dot)**: Represents that the number is included in the set. - **o (Open Circle)**: Represents that the number is **not** included in the set. - **— (Line)**: Indicates a range of numbers between two points. - **→ or ← (Arrow)**: Indicates that the set continues indefinitely in that direction. Feel free to ask if you need further clarification or visual examples!