Problem 49 - Algebra for College Students

Question

Express each set in simplest interval form. (Hint: Graph each set and look for the intersection or union.) $$ (-\infty,-6] \bigcap[-9, \infty) $$

Step-by-Step Solution

Answer

The simplest interval form is [-9, -6].

Step 1: Understand the intersection of intervals

The exercise requires finding the intersection of two intervals: o (-fty, -6] and o [-9, fty). For an intersection, we look for the common elements that are in both intervals.

Step 2: Graph the intervals

Graphically represent the two intervals on a number line.o (-fty, -6] is all numbers less than or equal to -6.o [-9, fty) is all numbers from -9 to positive infinity.

Step 3: Identify the overlap

Identify the common region where both intervals overlap. Since o (-fty, -6] starts before o [-9, fty) and overlaps until o -6, the intersection is the segment where both sets contain elements.

Step 4: Write the intersection

Combine the common regions to express the interval. The intersection of o (-fty, -6] and o [-9, fty) is o [-9, -6].

Key Concepts

intersection of intervals

When we talk about the intersection of intervals, we're looking for the set of numbers that are common to both intervals. Imagine two ranges on a number line. The intersection is where these ranges overlap.
In this exercise, we have two intervals: \((-\infty,-6]\) and \([-9, \infty)\). To find their intersection, we need to see where these intervals share common elements.
\((-\infty, -6]\) indicates all numbers less than or equal to -6. \([-9, \infty)\) includes all numbers from -9 to infinity.
By comparing these, we can see the common region is \([-9, -6]\). Both intervals include the numbers between -9 and -6.

interval notation

Interval notation is a way to describe a set of numbers along a number line. It uses brackets to indicate the endpoints of the intervals.
Here's how it works:

• Brackets like '[' and ']' mean the endpoint is included.
• Parentheses like '(' and ')' mean the endpoint is not included.

For example, \((-\infty, -6]\) means the interval starts from negative infinity (not included) and goes up to -6 (included). Similarly, \([-9, \infty)\) starts from -9 (included) and extends to positive infinity (not included).
When we write the intersection of these intervals, \([-9, -6]\), it shows the range from -9 to -6, inclusive of both endpoints.

number line

Graphing intervals on a number line can help visualize their relationship. A number line is a straight line where numbers are placed in their correct position.
Here’s how you can graph the intervals from the exercise:

• For \((-\infty, -6]\), shade the region from negative infinity up to and including -6.
• For \([-9, \infty)\), shade from -9 onwards to positive infinity.

Where the shading overlaps is the intersection. In this case, the overlap is from -9 to -6, which helps us decide the intersection is \([-9, -6]\).
This visual method makes understanding intervals much more accessible. It clearly shows where the intervals meet and how they relate to each other.