Problem 16 - Elementary and Intermediate Algebra

Question

a. Graph: \((-\infty, 2) \cup[3, \infty)\) b. Graph: \((-\infty, 3) \cap[-2, \infty)\)

Step-by-Step Solution

Answer

Graph a: Shade segments (-∞, 2) and [3, ∞); Graph b: Shade segment [-2, 3).

Step 1: Understand the Interval Notations

The interval notation \((-\infty, 2)\) refers to all real numbers less than 2. This interval is open because it does not include 2, and there is no upper bound (hence \(-\infty\)). \([3, \infty)\) refers to all real numbers greater than or equal to 3. This interval is closed at 3 because it includes 3, and it extends to infinity. The notation \((-\infty, 2) \cup [3, \infty)\) represents the union of these two intervals, meaning all numbers less than 2 or greater than or equal to 3.

Step 2: Graph \((-\infty, 2) \cup [3, \infty)\) on a number line

To graph \((-\infty, 2) \cup [3, \infty)\) on a number line, draw a line extending infinitely in both directions. Use an open circle at 2 and shade all numbers to the left to indicate that numbers less than 2 are included. From 3, use a closed circle (or filled dot) and shade to the right to indicate numbers from 3 onwards are included. Leave the section between 2 and 3 unshaded.

Step 3: Analyze Interval \((-\infty, 3) \cap [-2, \infty)\)

The interval \((-\infty, 3)\) includes all numbers less than 3. It is open because it doesn't include 3. The interval \([-2, \infty)\) includes all numbers starting at -2 and going to infinity. It is closed at -2 because it includes -2. The notation \((-\infty, 3) \cap [-2, \infty)\) represents the intersection of these two intervals, meaning all numbers that are common to both intervals.

Step 4: Graph \((-\infty, 3) \cap [-2, \infty)\) on a Number Line

To graph \((-\infty, 3) \cap [-2, \infty)\), we are looking for numbers both less than 3 and also greater than or equal to -2. So, on a number line, start at -2 with a closed circle, and shade the region up to, but not including, 3. Use an open circle at 3. This shows all numbers in the range [-2, 3) are included in the intersection.

Key Concepts

Graphing Intervals

Graphing intervals is an important skill to visualize the range of numbers that satisfy certain conditions. To graph an interval, the number line is used as our basic tool. When graphing

• Open Intervals: These exclude the endpoint(s) indicated by parentheses. An open circle is placed at the endpoint, such as in \((-\infty, 2)\), which includes all numbers less than 2 but not 2 itself.
• Closed Intervals: These include the endpoint, indicated by square brackets. A filled circle is used, as with \([3, \infty)\), which includes the number 3 and anything greater.
• Union of Intervals: \((a, b) \cup [c, d)\) combines intervals, meaning numbers are included in either range.
• Intersection of Intervals: \((a, b) \cap [c, d)\) includes only numbers present in both intervals at once.

Using these principles, you can represent complicated inequalities simply by shading parts of a number line between these points.

Number Line

The number line is a visual representation of all real numbers arranged in order. It helps in understanding the concept of intervals by providing a straightforward illustration:

• Basic Structure: The line extends infinitely in both directions, accommodating all numbers.
• Points on the Line: Specific points are marked to represent numbers, such as integers or important fractions.
• Direction and Order: Numbers to the right are greater, while those to the left are smaller.
• Interval Representation: Shading or circling portions shows which numbers are included in a given interval.

By effectively using a number line, intervals become more tangible and much easier to grasp. It turns abstract numerical concepts into a more concrete and visual form.

Set Notation

Set notation is a mathematical shorthand used to describe a collection of numbers or objects. It's essential for expressing the solutions to equations or inequalities, and understanding how intervals fit into this framework:

• Elements: Items within the set, represented typically by numbers.
• Set Symbols: Braces \(\{ \}\) are used to denote a set, and elements are placed inside.
• Interval Relations: Just as intervals can describe elements graphically, set notation describes them algebraically. For example, the union \(A \cup B\) includes all elements in either set A or B.
• Intersection: This is denoted by \(A \cap B\), including elements common to both sets.

Using set notation keeps mathematics precise yet concise, especially when dealing with extensive or complex groups of numbers.