Question

Use inequality notation to describe the subset of real numbers.\(y\) is greater than 5 and less than or equal to 12 .

Short answer

The subset of real numbers where \(y\) is greater than 5 and less than or equal to 12 is represented by the inequality \(5 < y \leq 12\).

Step-by-step solution

Identify the bounds

From the given exercise, it's understood that \(y\) is more than 5 and less than or equal to 12.

Write the inequality for \(y\) is greater than 5

To represent that \(y\) is greater than 5, an inequality can be written as \(y > 5\). This means that \(y\) can be any real number that is more than 5.

Write the inequality for \(y\) is less than or equal to 12

To represent that \(y\) is less than or equal to 12, the inequality can be written as \(y \leq 12\). This means that \(y\) can be any real number that is less than or equal to 12.

Combine the two inequalities

Both the inequalities can be combined into one to represent that \(y\) is more than 5 and at the same time less than or equal to 12. The combined notation will be \(5 < y \leq 12\).

Key concepts

Real Numbers

Real numbers are the set of numbers that include both rational numbers, like integers and fractions, and irrational numbers, such as \sqrt{2}\ or \pi\. The concept of real numbers is foundational in mathematics, serving as the broadest context for numbers used in everyday calculations and scientific measurements. Some of the critical properties of real numbers are:

• They include whole numbers, fractions, and decimals.
• They extend infinitely in both the positive and negative directions.
• They can be plotted on a number line, where every point represents a real number.
• They are dense, meaning between any two real numbers, there is always another real number.

In the exercise, we are dealing with real numbers when describing possible values of \(y\). This means \(y\) can take any value on the number line between specific bounds, subject to the inequalities stated.

Inequalities

An inequality is a mathematical statement that tells us how two values relate to each other. Unlike equations that declare two expressions are equal, inequalities express one quantity being greater or lesser than another. There are various notations in inequalities, such as:

• \(<\) means less than.
• \(>\) means greater than.
• \(\leq\) means less than or equal to.
• \(\geq\) means greater than or equal to.

In solving inequalities, such as in the given exercise, the requirement was to express \(y\) as being greater than 5 and less than or equal to 12. This can be done by writing two separate inequalities: \(y > 5\) and \(y \leq 12\), and then combining them to form the compound inequality \(5 < y \leq 12\). Understanding inequalities is crucial for describing ranges or intervals in real numbers.

Bounds and Intervals

Bounds and intervals are ways to precisely describe the range of values that a variable can take, especially in the context of real numbers. Bounds:

• These are the limits or endpoints of an interval. In the expression \(5 < y \leq 12\), 5 and 12 are bounds.
• The lower bound, 5, is not included in the set of values, hence the use of the open inequality \(>\).
• The upper bound, 12, is included in the values, shown by the closed inequality \(\leq\).

Intervals:

• Described using the bounds, they specify which real numbers are included. Here, we are dealing with a half-open interval.
• A half-open interval includes one of the endpoint values but not the other. In \(5 < y \leq 12\), the interval contains all numbers \(y\) that are between and equal to 12 but greater than 5.
• Such intervals can be represented on a number line by drawing a line between the bounds, using open or closed circles to indicate whether bounds are included or not.

The concept of intervals is central to many areas of mathematics, helping to delineate sections of the numerical landscape for study or application.