(d) Represent the following sets on a number line: (1) \( \{x:-3

Ask by Craig Martinez. in South Africa

Jan 26,2025

Certainly! Let's represent each of the given sets on a number line. Since I can't provide actual drawings, I'll describe how each set would appear graphically on a number line. For clarity, open circles (○) indicate that a boundary is not included, and closed circles (●) indicate that a boundary is included. Shaded regions represent the range of numbers in the set. Additionally, integers (\(\mathbb{Z}\)) will be represented as distinct points, while real numbers (\(\mathbb{R}\)) will be shown as continuous intervals. ### (1) \( \{x : -3 < x < 7 ;\ x \in \mathbb{Z}\} \) **Description:** - **Type:** Integers between -3 and 7, not including -3 and 7. - **Elements:** \(-2, -1, 0, 1, 2, 3, 4, 5, 6\). **Number Line Representation:** ``` ---●--○--●--●--●--●--●--●--●--●--○--- -3 -2 -1 0 1 2 3 4 5 6 7 ``` - **Open circles** at -3 and 7 indicate these values are not included. - **Filled circles** at each integer from -2 to 6 represent the elements of the set. --- ### (2) \( \{x : x \geq -5 ;\ x \in \mathbb{R}\} \) **Description:** - **Type:** All real numbers greater than or equal to -5. - **Range:** \([-5, \infty)\). **Number Line Representation:** ``` ●----------------------------→ -5 ∞ ``` - **Closed circle** at -5 indicates that -5 is included. - **Arrow** pointing to the right signifies all numbers greater than -5 are included. --- ### (3) \( \{x : -2 \leq x < 4 ;\ x \in \mathbb{R}\} \) **Description:** - **Type:** All real numbers between -2 and 4, including -2 but not 4. - **Range:** \([-2, 4)\). **Number Line Representation:** ``` ●====================○→ -2 4 ``` - **Closed circle** at -2 indicates inclusion. - **Open circle** at 4 indicates exclusion. - **Shaded line** between -2 and 4 represents all real numbers in this interval. --- ### (4) \( \{x : x < 3 ;\ x \in \mathbb{Z}\} \) **Description:** - **Type:** All integers less than 3. - **Elements:** \( \ldots, -3, -2, -1, 0, 1, 2 \). **Number Line Representation:** ``` ←---○--●--●--●--●--●--● ... -3 -2 -1 0 1 2 3 ``` - **Open circle** at 3 indicates exclusion. - **Filled circles** at each integer less than 3 represent the elements. - **Arrow** pointing to the left shows that the set extends infinitely in the negative direction. --- ### (5) \( \{x : x \leq 6.5 ;\ x \in \mathbb{R}\} \) **Description:** - **Type:** All real numbers less than or equal to 6.5. - **Range:** \((-\infty, 6.5]\). **Number Line Representation:** ``` ←====================● 6.5 ``` - **Closed circle** at 6.5 indicates inclusion. - **Shaded line** extending to the left represents all real numbers less than or equal to 6.5. --- ### (6) \( \left\{x : x \geq 2 \frac{1}{2} ;\ x \in \mathbb{R}\right\} \) **Description:** - **Type:** All real numbers greater than or equal to 2.5. - **Range:** \([2.5, \infty)\). **Number Line Representation:** ``` ●----------------------------→ 2.5 ∞ ``` - **Closed circle** at 2.5 indicates inclusion. - **Arrow** pointing to the right signifies all numbers greater than or equal to 2.5 are included. --- ### (7) \( \{x : x < -3 ;\ x \in \mathbb{Z}\} \) **Description:** - **Type:** All integers less than -3. - **Elements:** \( \ldots, -5, -4\). **Number Line Representation:** ``` ←---○--●--●--● ... -5 -4 -3 ``` - **Open circle** at -3 indicates exclusion. - **Filled circles** at each integer less than -3 represent the elements. - **Arrow** pointing to the left shows that the set extends infinitely in the negative direction. --- ### (8) \( \{x : 0 \leq x < 5 ;\ x \in \mathbb{R}\} \) **Description:** - **Type:** All real numbers between 0 and 5, including 0 but not 5. - **Range:** \([0, 5)\). **Number Line Representation:** ``` ●====================○→ 0 5 ``` - **Closed circle** at 0 indicates inclusion. - **Open circle** at 5 indicates exclusion. - **Shaded line** between 0 and 5 represents all real numbers in this interval. --- ### Summary - **Closed Circle (●):** The boundary point is included in the set. - **Open Circle (○):** The boundary point is not included. - **Shaded Line (=====):** Represents all real numbers within the interval. - **Filled Circles for \(\mathbb{Z}\):** Represents each integer in the set. - **Arrows (→ or ←):** Indicate the direction in which the set extends infinitely. By following these representations, you can accurately depict each set on a number line.