Problem 47 - Algebra for College Students

Question

Express each set in simplest interval form. (Hint: Graph each set and look for the intersection or union.) $$ (-\infty,-1] \cap[-4, \infty) $$

Step-by-Step Solution

Answer

The intersection is \([-4, -1]\).

Step 1: Understand the given sets

The problem involves two sets: 1. \((-\infty, -1]\), which includes all numbers from negative infinity to -1, including -1.2. \([-4, \infty)\), which includes all numbers from -4 to positive infinity, including -4.

Step 2: Graph the sets

Graph each set on a number line: - For \((-\infty, -1]\), draw a line from negative infinity to -1, including -1 (solid dot on -1).- For \([-4, \infty)\), draw a line from -4 to positive infinity, including -4 (solid dot on -4).

Step 3: Find the intersection

The intersection (\(\textbackslash\textbackslashcap\)) of these sets is where the two lines overlap. The overlap starts at -4 and ends at -1. Since -1 is included in both sets and -4 is also included, the intersection is the interval that includes both endpoints.

Step 4: Write the intersection in interval form

The intersection of \((-\infty, -1]\) and \([-4, \infty)\) is \([-4, -1]\).

Key Concepts

Set Intersection

The concept of 'set intersection' is fundamental in algebra. When you take the intersection of two sets, you're finding the elements that both sets have in common. For example, consider the sets \( (-\infty,-1] \) and \([-4, \infty)\):

* Set \( (-\infty, -1] \) includes all numbers from negative infinity to -1 (including -1 itself). * Set \([-4, \infty)\) includes all numbers from -4 to positive infinity (including -4 itself).

To find the intersection of these two sets, you look for where they overlap. Graphically, these sets intersect between -4 and -1.
This means that every number from -4 to -1 (including both -4 and -1) is part of the intersection.

In mathematical notation, we write the intersection as \([-4, -1]\). This tells us that the only numbers that belong to both sets at the same time are those between -4 and -1.

Graphing Intervals

Graphing intervals on a number line is a handy visual method to understand where two sets intersect or combine. To graph an interval:
1. Mark the endpoints of the interval on the number line.
2. Use a solid dot to include the endpoint in the interval, or an open dot to exclude it.
3. Draw a line between these points to show the range of numbers included.

Let's graph our example sets:

* For \( (-\infty, -1] \):
- Start from negative infinity and go up to -1
- Include -1 by drawing a solid dot

* For \([-4, \infty)\):
- Start at -4 with a solid dot
- Go to positive infinity

When graphing these on the same number line, the overlap or intersection is clearly visible between -4 and -1. This is because both sets include numbers from -4 to -1.

Interval Notation in Algebra

Interval notation in algebra is a standard way of representing a range of numbers. We use brackets and parentheses to indicate which numbers are included or excluded in an interval.

* A square bracket \[ \] means the endpoint is included (closed interval).
* A parenthesis \( \) means the endpoint is excluded (open interval).

In our example, we encountered two sets:
* \( (-\infty, -1] \)
* \([-4, \infty)\)

The intersection of these sets is represented as \([-4, -1]\).
* Here, \[ -4 \] indicates that -4 is included.
* \[ -1 \] signifies that -1 is included as well.

Understanding interval notation allows you to succinctly describe a wide variety of sets and ranges. This is a crucial skill in algebra, enabling you to solve and interpret problems effectively.