Problem 150 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement . $$(-\infty,-1] \cap[-4, \infty)=[-4,-1]$$

Step-by-Step Solution

Answer

The statement is true. The intersection of the intervals \((-\infty,-1]\) and \([-4, \infty)\) is indeed \([-4,-1]\).

Step 1: Interpret the intervals

The first interval \((-\infty,-1]\) denotes all numbers less than or equal to -1. The second interval \([-4, \infty)\) denotes all numbers greater than or equal to -4.

Step 2: Identify the common elements

To find the intersection, we identify the common elements in the two intervals. These are all numbers from -4 to -1 inclusive.

Step 3: Formulate the intersection

Formulate the intersection as a new interval. The correct expression for the intersection is \([-4,-1]\).

Key Concepts

Interval Notation

Interval notation is a way of representing sets of numbers, specifically what we call intervals. Think of an interval as a chunk of numbers on the number line.
It shows where an interval starts and ends. Let's break it down:

• The interval \((-\infty, -1]\) includes all numbers less than or equal to -1. The square bracket \([-1]\) signifies that -1 is included in this interval, while the parenthesis \((-\infty)\) means that the interval extends infinitely in the negative direction but does not include infinity itself.


• For the interval \([-4, \infty)\), it contains numbers greater than or equal to -4. The square bracket \([-4]\) again shows -4 is included, and the parenthesis \(\infty)\) indicates the interval stretches indefinitely towards positive infinity but doesn't include infinity.

In general, use square brackets \([\text{ or } ]\) for inclusive bounds, and parentheses \((\text{ or } )\) for exclusive bounds.

Intersection of Sets

The intersection of sets, in simple terms, is all about finding what is common between two sets. In mathematical language, when we intersect two sets, we review them to spot elements they share.
Here's how it applies to intervals:

• The first set here is \((-\infty, -1]\), meaning it includes numbers up to and including -1.


• The second set is \([-4, \infty)\), consisting of numbers from -4 upwards.


• When we look for common numbers (intersection), we're checking what numbers exist in both intervals simultaneously.

In the example exercise, the overlap (or intersection) is \([-4, -1]\). These numbers fall within both original intervals.

Inequality Notation

Inequality notation is another way to describe a range of numbers or a particular set of values. Instead of using brackets, it uses inequality symbols to capture a range.
For instance, \( x \leq -1 \) describes all numbers less than or equal to -1. Here's a quick look:

• The expression \( x \leq -1 \) translates to the interval \((-\infty, -1]\), which includes everything up to -1.


• Similarly, \( x \geq -4 \) represents numbers starting from -4 and greater, just like interval \([-4, \infty)\).

Both inequality and interval notations are useful for illustrating intervals and make it easier to understand the set operations like intersection or union.