Problem 52 - Algebra and Trigonometry

Question

Graph the set. $$ (-\infty, 6] \cap(2,10) $$

Step-by-Step Solution

Answer

The set \((-\infty, 6] \cap (2, 10)\) is \((2, 6]\).

Step 1: Understanding Interval Notations

The notation \((-\infty, 6]\) represents all numbers less than or equal to 6. \((2, 10)\) represents numbers greater than 2 and less than 10. We are finding the intersection, meaning values that exist in both intervals.

Step 2: Identifying the Intersection

To find the intersection, we need numbers common to both intervals. For \((-\infty, 6]\): all numbers up to and including 6.For \((2, 10)\): all numbers greater than 2 and less than 10.Thus, the intersection is all numbers greater than 2 and up to 6.

Step 3: Write the Intersection as an Interval

The set of numbers that satisfy both conditions is represented as the interval \((2, 6]\). This includes 3, 4, 5 and 6, but not 2 because \((2, 10)\) excludes 2.

Step 4: Graph the Interval

To graph the interval \((2, 6]\),- Place an open circle on 2 (indicating that 2 is not included).- Place a closed circle on 6 (indicating that 6 is included).- Shade the region on the number line between 2 and 6, to represent all numbers in that range.

Key Concepts

Interval Intersection

When learning about interval intersection, it's all about finding numbers that belong to both sets of numbers, or intervals. Intervals can be thought of as a range of numbers represented within certain boundaries.
For example, if you have

• Interval A:
  (−∞, 6]
• Interval B:
  (2, 10)

These can be interpreted as follows:

• 
  Interval A includes all numbers less than or equal to 6.
• Interval B includes numbers greater than 2 and less than 10.

When we talk about the intersection, we're looking for all the numbers that these two intervals have in common. This involves a little bit of overlap to see which numbers are part of both intervals at the same time. In this case, the intersection of

• 
  (−∞, 6]
• (2, 10)

is the interval

• (2, 6]

which includes numbers greater than 2 but less than or equal to 6.

Graphing Intervals

Graphing intervals is a way of visualizing ranges on a number line, and it can help in understanding the scope of numbers involved in an interval. This method provides a clear picture of what the interval represents.
For graphing the interval

• (2, 6]

follow these steps:

• Draw a number line and mark points representing the numbers 2 and 6.
• Place an open circle on the number 2. This suggests that 2 is not part of the interval and is just a boundary. The open circle reminds us that the value right at that point isn't included.
• Place a closed circle on the number 6. The closed circle indicates inclusivity, meaning 6 is included in the interval.
• Shade the area on the number line that lies between the open circle at 2 and the closed circle at 6.

This shading visually shows all numbers that are part of the interval (2, 6). It tells us which numbers satisfy the condition of being greater than 2 but less than or equal to 6.

Number Line Graphing

Number line graphing is an essential skill for visually conveying mathematical intervals. A number line is a straight line with numbers placed at equal intervals along it, and it is particularly useful in displaying numerical operations and relationships.
When graphing intervals using a number line, follow these tips:

• Select the section of the number line relevant to your intervals. You don't need to draw the entire line, just the portion relevant to your interval, such as from 0 to 10 in our example.
• Identify your interval boundaries on the number line. For
  example, for (2, 6], you would highlight points at 2 and 6.
• Use open and closed circles to show the boundaries. Open circles for values that aren't included and closed for those that are.
• Using shading between these circles helps quickly interpret which numbers are part of the interval.

Using a number line to graph intervals simplifies understanding complex mathematical intervals by providing a straightforward visual aid. It's especially helpful in recognizing how intervals relate to each other, such as when interpreting intersections, differences, or unions of multiple intervals.