Help me understand and solve interval notation and set notation problems---**KEY IDEA** Unbounded Intervals on the Real Number Line Let a and b be real numbers. Each interval on the real number line shown below is called an **unbounded interval**. | Inequality | Interval Notation | Graph | | :--------- | :---------------- | :------------------------------------------------------------------------ | | x ≥ a | [a, ∞) | Number line with 'a' marked, a solid dot at 'a', and an arrow extending to the right from 'a'. The axis is labeled 'x'. | | x > a | (a, ∞) | Number line with 'a' marked, an open circle at 'a', and an arrow extending to the right from 'a'. The axis is labeled 'x'. | | x ≤ b | (-∞, b] | Number line with 'b' marked, a solid dot at 'b', and an arrow extending to the left from 'b'. The axis is labeled 'x'. | | x < b | (-∞, b) | Number line with 'b' marked, an open circle at 'b', and an arrow extending to the left from 'b'. The axis is labeled 'x'. | | | (-∞, ∞) | Number line with arrows extending indefinitely to both the left and right. The axis is labeled 'x'. | The symbols ∞ (infinity) and -∞ (negative infinity) are used to represent the unboundedness of intervals such as [7, ∞) and (-∞, 7]. Because these symbols do not represent real numbers, they are always enclosed by a parenthesis. **EXAMPLE 1** Writing Interval Notation Write each interval in interval notation. a. -2 ≤ x ≤ 3 b. x > -1 c. **Chart Description:** * **Type:** Number line graph. * **Main Elements:** * **X-axis:** Labeled 'x', with tick marks and integer labels from -5 to 5. * **Points:** A solid dot is located at -3. An open circle (unfilled) is located at 4. * **Line:** A thick, solid line segment connects the solid dot at -3 to the open circle at 4, indicating all numbers between -3 and 4 (inclusive of -3, exclusive of 4). * **Direction:** The number line itself has arrows on both ends. d. **Chart Description:** * **Type:** Number line graph. * **Main Elements:** * **X-axis:** Labeled 'x', with tick marks and integer labels from -5 to 5. * **Points:** A solid dot is located at 3. * **Line:** A thick, solid line starts at the solid dot at 3 and extends indefinitely to the left (towards negative infinity), ending with an arrow. * **Direction:** The number line itself has arrows on both ends. **SOLUTION** a. The graph of -2 ≤ x ≤ 3 is the bounded interval [-2, 3]. b. The graph of x > -1 is the unbounded interval (-1, ∞). c. The graph represents all the real numbers between -3 and 4, including the endpoint -3. This is the bounded interval [-3, 4). d. The graph represents all the real numbers less than or equal to 3. This is the unbounded interval (-∞, 3]. **SELF-ASSESSMENT** 1 I do not understand. 2 I can do it with help. 3 I can do it on my own. 4 I can teach someone else. Write the interval in interval notation. 1. -7 < x < -4 2. x ≤ 5 3. **Chart Description:** * **Type:** Number line graph. * **Main Elements:** * **X-axis:** Labeled 'x', with tick marks and integer labels from -5 to 5. * **Points:** A solid dot is located at -2. * **Line:** A thick, solid line starts at the solid dot at -2 and extends indefinitely to the right (towards positive infinity), ending with an arrow. * **Direction:** The number line itself has arrows on both ends. **Page Footer:** 1.1 Interval Notation and Set Notation 5 Here is the complete and accurate extraction of all content from the image in a structured plain text format: --- **Page Header** Using Interval Notation **Vocabulary Box** **Vocabulary** set, p. 4 subset, p. 4 endpoints, p. 4 bounded interval, p. 4 unbounded interval, p. 5 set-builder notation, p. 6 **Main Text - Set Notation** In mathematics, a collection of objects is called a **set**. You can use braces { } to represent a set by listing its members or **elements**. For instance, the set {1, 2, 3} A set with three members contains the three numbers 1, 2, and 3. A set with no elements, the **empty set** (or **null set**), can be represented by empty braces, or with the symbol Ø. Many other sets are also described in words, such as the set of real numbers. If all the members of a set A are also members of a set B, then set A is a **subset** of set B. The set of natural numbers {1, 2, 3, 4, ...} is a subset of the set of real numbers. The diagram shows several important subsets of the real numbers. **READING Box** The symbols represent subsets of the real numbers. R: Real numbers Q: Rational numbers Z: Integers W: Whole numbers N: Natural numbers **Chart Description - Number Set Diagram** * **Type:** Nested rectangle diagram with an adjacent rectangle, representing relationships between sets of numbers. * **Main Elements:** * **Outer Rectangle:** Labeled "Real Numbers (R)". * **Nested Rectangles (from largest to smallest, inside "Real Numbers"):** * "Rational Numbers (Q)" * "Integers (Z)" * "Whole Numbers (W)" * "Natural Numbers (N)" * **Adjacent Rectangle:** Located to the right of the nested rectangles, labeled "Irrational Numbers". * **Relative Position:** The diagram illustrates that Natural Numbers are a subset of Whole Numbers, which are a subset of Integers, which are a subset of Rational Numbers. Rational Numbers and Irrational Numbers are both subsets of Real Numbers, and they are distinct (disjoint) from each other. **Connecting Sentence** Many subsets of the real numbers can be represented as **intervals** on the real number line. **KEY IDEA Section** **KEY IDEA** **Bounded Intervals on the Real Number Line** Let *a* and *b* be two real numbers such that *a* < *b*. Then *a* and *b* are the **endpoints** of four different **bounded intervals** on the real number line, as shown below. A bracket or closed circle indicates that the endpoint is included in the interval and a parenthesis or open circle indicates that the endpoint is not included in the interval. **Table Content - Bounded Intervals** | Inequality | Interval Notation | Graph | | :--------- | :---------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | a ≤ x ≤ b | [a, b] | A horizontal number line with arrows on both ends. A solid line segment connects point 'a' and point 'b'. Point 'a' has a closed (filled) circle. Point 'b' has a closed (filled) circle. The label 'x' is above the line segment. | | a < x < b | (a, b) | A horizontal number line with arrows on both ends. A solid line segment connects point 'a' and point 'b'. Point 'a' has an open (unfilled) circle. Point 'b' has an open (unfilled) circle. The label 'x' is above the line segment. | | a ≤ x < b | [a, b) | A horizontal number line with arrows on both ends. A solid line segment connects point 'a' and point 'b'. Point 'a' has a closed (filled) circle. Point 'b' has an open (unfilled) circle. The label 'x' is above the line segment. | | a < x ≤ b | (a, b] | A horizontal number line with arrows on both ends. A solid line segment connects point 'a' and point 'b'. Point 'a' has an open (unfilled) circle. Point 'b' has a closed (filled) circle. The label 'x' is above the line segment. | **Text Below Table** The length of any bounded interval, [a, b], (a, b), [a, b), or (a, b], is the distance between its endpoints: *b* - *a*. Any bounded interval has a *finite* length. An interval that does not have a finite length is called **unbounded** or **infinite**. **Page Footer** 4 Chapter 1 Linear Functions, Linear Systems, and Matrices --- Here is the complete and accurate extraction of the content from the image: --- **Page Header:** 1.1 Interval Notation and Set Notation GO DIGITAL (with QR code symbol) **Learning Target:** Use interval notation and set-builder notation. **Success Criteria:** * I can represent intervals using interval notation. * I can represent intervals using set-builder notation. **EXPLORE IT! Writing Subsets in Set Notation** Work with a partner. A collection of objects is called a **set**. You can use braces { } to represent a set by listing its members or by using **set-builder notation** to define the set in terms of the properties of its members. For instance, the set of the numbers 1, 2, and 3 can be denoted as {1, 2, 3} *Annotation:* List the members of the set in braces. and the set of all odd whole numbers can be denoted as {x | x is a whole number and x is odd} *Annotation:* Set-builder notation which is read "The set of all real numbers x such that x is a whole number and x is odd." If all of the members of a set A are also members of a set B, then set A is a **subset** of set B. For instance, if set A = {a, b} and set B = {a, b, c, d}, then set A is a subset of set B. **Practice Questions:** **a. Question Stem:** Write all the nonempty subsets of each set. i. {4, 5} ii. {c, d} iii. {2, 4, 6} iv. {e, f, g, h} **b. Question Stem:** Write each given subset of the real numbers in set-builder notation. i. the integers ii. the whole numbers iii. the natural numbers iv. the rational numbers v. the irrational numbers vi. the positive integers **c. Question Stem:** Write each indicated set of numbers using either braces to list its members or set-builder notation. Explain your choice of notation. i. the whole numbers 50 through 54 ii. the real numbers 0 through 4 iii. the prime whole numbers iv. the integers -100 through 100 **Analyze Relationships Box:** Describe any relationships among these subsets of the real numbers. **Page Footer:** 1.1 Interval Notation and Set Notation 3 **Chart/Diagram Description:** * **Type:** Photographic image of a person. * **Main Elements:** The image displays a young man, appearing to be a student. He is smiling and looking directly at the viewer. He has short dark hair and is wearing a light blue denim jacket over a grey hooded sweatshirt. A blue backpack is visible on his back, and he is holding a green folder or book in his left arm. Light blue headphones are worn around his neck. * **Relative Position:** The figure of the young man is positioned on the left side of the page, occupying a significant portion of its height and width, approximately half of the page's vertical extent. --- Here is the complete and accurate extraction of all content from the image: **Title:** Using Set-Builder Notation **Introduction:** Another way to represent intervals is to write them in set-builder notation. --- **KEY IDEA** **Set-Builder Notation** Set-builder notation uses symbols to define a set in terms of the properties of the members of the set. | Set-builder notation | Words | Graph --- **EXAMPLE 2 Using Set-Builder Notation** Sketch the graph of each set of numbers. a. {x | 2 < x ≤ 5} b. {x | x ≤ 0 or x > 4} **SOLUTION** a. The real numbers in the set satisfy both x > 2 and x ≤ 5. **Chart/Diagram Description:** * **Type:** Number line. * **Coordinate Axes:** A horizontal axis labeled 'x' with arrows on both ends. * **Scale:** Integer marks from -1 to 6 are explicitly labeled. * **Points:** An open circle is located at 2. A closed circle (filled) is located at 5. * **Line:** The segment of the number line between 2 and 5 is shaded, connecting the open circle at 2 and the closed circle at 5. b. The real numbers in the set satisfy either x ≤ 0 or x > 4. **Chart/Diagram Description:** * **Type:** Number line. * **Coordinate Axes:** A horizontal axis labeled 'x' with arrows on both ends. * **Scale:** Integer marks from -2 to 5 are explicitly labeled. * **Points:** A closed circle (filled) is located at 0. An open circle is located at 4. * **Lines:** * A shaded line segment extends from the closed circle at 0 indefinitely to the left, indicated by an arrow. * A shaded line segment extends from the open circle at 4 indefinitely to the right, indicated by an arrow. --- **WORDS AND MATH** The symbol ∈ denotes membership in a set. The expression x ∈ Z means that x is a member (or element) of the set of integers. --- **EXAMPLE 3 Writing Set-Builder Notation** Write the set of numbers in set-builder notation. a. the set of all integers greater than 5 b. (-∞, -1) or (-1, ∞) **SOLUTION** a. x is greater than 5 and x is an integer. {x | x > 5 and x ∈ Z} b. x can be any real number except -1. {x | x ≠ -1} --- **SELF-ASSESSMENT** [Buttons: 1 I do not understand. 2 I can do it with help. 3 I can do it on my own. 4 I can teach someone else.] Sketch the graph of the set of numbers. 4. -6 < x ≤ -2 5. {x | x ≤ 0 or x ≥ 10} Write the set of numbers in set-builder notation. 6. (-∞, -1] or (1, ∞) 7. The set of all integers except -4