Problem 36 - Calculus: One and Several Variables

Question

Sketch the set on a number line. (-4,3].

Step-by-Step Solution

Answer

In the number line, an open circle is at -4 indicating the set does not include -4. A closed circle is drawn at 3 along with the shaded region between -4 and 3, indicating that the set includes all the numbers between and including 3. All other parts of the line remain unshaded.

Step 1 Title

Understand the notations. (-4,3] is an interval notation in which the parentheses '()' symbolizes that the number is not included in the set and the square brackets '[]' means the number is included. Therefore, the given set includes the numbers greater than -4 and lesser than or equal to 3.

Step 2 Title

Sketch a number line. It should be long enough to include the numbers specified in your set. Then, mark off equal distances and mark each with the appropriate integer (for example, -4, -3, -2, -1, 0, 1, 2, 3).

Step 3 Title

Indicate the appropriate range on the number line. Because -4 is not included, draw an open circle at -4. Because 3 is included, draw a closed circle at 3. Finally, shade the numbers between -4 and 3 to indicate that all of these numbers are included in the set.

Key Concepts

Number Line Overview

A number line is a visual tool that represents numbers in a straight line. It's helpful for understanding the relationship between numbers and can show complex mathematical concepts simply.

On a number line, numbers are placed at equal intervals or distances from each other. The center is usually labeled with zero. Negative numbers are typically shown to the left and positive numbers to the right.

When sketching a number line:

• Start by drawing a horizontal line with arrows on both ends. This indicates that the line extends infinitely in both directions.
• Mark equal intervals along the line, and label these points with key integers, such as -4, -3, 0, 1, 2, and so on.
• This layout helps in visualizing operations like addition, subtraction, and locating numbers within intervals.

Open and Closed Intervals Explained

Intervals describe a range of numbers between two endpoints on a number line. They can be open, closed, or half-open (mixed), depending on whether the endpoints are included in the interval.

• Open Interval (a, b): This interval does not include the endpoints. In an open interval, we use parentheses. For example, in \((-4, 3)\), -4 and 3 are not included.
• Closed Interval [a, b]: This interval includes the endpoints. We use square brackets to show this. For example, in \([-4, 3]\), both -4 and 3 are included.
• Half-Open Interval (a, b] or [a, b): This interval includes one endpoint but not the other. For instance, in \((-4, 3]\), -4 is not included, but 3 is.

When visualizing these on a number line, use an open circle for non-included endpoints and a closed circle (filled dot) for included ones. This helps in clearly identifying which numbers are part of the interval.

Understanding Set Notation

Set notation is a method used to define the elements within a set. It is efficient for expressing collections of numbers, such as those seen in intervals.

Using set notation makes it easy to understand exactly which numbers are included:

• An interval such as \((-4, 3]\) is a classic example. Here, it includes all real numbers between -4 and 3, except -4 itself but including 3.
• Set notation can also be represented as \( \{ x \mid -4 < x \leq 3 \} \). This means "all numbers \(x\) such that \(-4 < x \leq 3\).
• This form of notation helps in both mathematical reasoning and problem-solving, providing clarity on which numbers belong to the set.

Understanding set notation ensures students can interpret and communicate the specific bounds and continuity of number sets effectively.