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Draw a number line for each of the following: a) хе(-6; 2) b) {x E R:x > 3} c) y EZ and [-3; 7) d) {x € N: x 2 -2} e) (x E R: -4 < x< 5}

Draw a number line for each of the following: a) хе(-6; 2) b) {x E R:x > 3} c) y EZ and [-3; 7) d) {x € N: x 2 -2} e) (x E R: -4 < x< 5}

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Number Line Visualizations To enhance understanding of set notation and intervals, draw a number line for each of the following cases: **a) \( x \in (-6, 2) \)** This represents all real numbers \( x \) between -6 and 2, not including -6 and 2. On the number line, this is indicated by an interval starting just after -6 and ending just before 2. **b) \( \{ x \in R : x > 3 \} \)** This represents all real numbers \( x \) that are greater than 3. On the number line, this starts just after 3 and extends infinitely to the right. **c) \( y \in Z \text{ and } [-3, 7) \)** This represents all integers \( y \) within the closed interval from -3 to 7, not including 7. On the number line, this includes -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6. **d) \( \{ x \in N : x \ge -2 \} \)** This represents all natural numbers \( x \) (which are typically positive whole numbers starting from 1). Since the interval \( x \ge -2 \) refers to values greater than or equal to -2, but natural numbers start from 1, this interval does **not have valid natural numbers** because natural numbers do not include negative numbers or zero. **e) \( \{ x \in R : -4 < x \le 5 \} \)** This represents all real numbers \( x \) which are greater than -4 and less than or equal to 5. On the number line, this starts just after -4 and ends at the value of 5 including 5. ### Visualization Explanation To visualize each interval: 1. **Identify the interval limits (both open and closed) on the number line.** 2. **Use an open circle to indicate that a number is not included (for example, -6 in case a).** 3. **Use a closed circle to indicate that a number is included (for example, 5 in case e).** 4. **Shade the region between the circles to indicate all the numbers included in the interval (for example, between -6 and 2 in case
Transcribed Image Text:### Number Line Visualizations To enhance understanding of set notation and intervals, draw a number line for each of the following cases: **a) \( x \in (-6, 2) \)** This represents all real numbers \( x \) between -6 and 2, not including -6 and 2. On the number line, this is indicated by an interval starting just after -6 and ending just before 2. **b) \( \{ x \in R : x > 3 \} \)** This represents all real numbers \( x \) that are greater than 3. On the number line, this starts just after 3 and extends infinitely to the right. **c) \( y \in Z \text{ and } [-3, 7) \)** This represents all integers \( y \) within the closed interval from -3 to 7, not including 7. On the number line, this includes -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6. **d) \( \{ x \in N : x \ge -2 \} \)** This represents all natural numbers \( x \) (which are typically positive whole numbers starting from 1). Since the interval \( x \ge -2 \) refers to values greater than or equal to -2, but natural numbers start from 1, this interval does **not have valid natural numbers** because natural numbers do not include negative numbers or zero. **e) \( \{ x \in R : -4 < x \le 5 \} \)** This represents all real numbers \( x \) which are greater than -4 and less than or equal to 5. On the number line, this starts just after -4 and ends at the value of 5 including 5. ### Visualization Explanation To visualize each interval: 1. **Identify the interval limits (both open and closed) on the number line.** 2. **Use an open circle to indicate that a number is not included (for example, -6 in case a).** 3. **Use a closed circle to indicate that a number is included (for example, 5 in case e).** 4. **Shade the region between the circles to indicate all the numbers included in the interval (for example, between -6 and 2 in case
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