Problem 52 - Algebra and Trigonometry

Question

\(47-52\) : Express the inequality in interval notation, and then graph the corresponding interval. $$ -5

Step-by-Step Solution

Answer

Interval: \((-5, 2)\); graph is a number line with open circles at \(-5\) and \(2\).

Step 1: Understand the Inequality

The given inequality is \(-5 < x < 2\). It means that \(x\) is greater than \(-5\) and less than \(2\). This defines a range of values for \(x\).

Step 2: Express in Interval Notation

We express the range of values in interval notation. The interval \((-5, 2)\) represents all numbers between \(-5\) and \(2\), not including \(-5\) and \(2\) themselves.

Step 3: Graph the Interval

To graph \((-5, 2)\), draw a number line. Place an open circle at \(-5\) and another open circle at \(2\) to show that these endpoints are not included in the interval. Shade the region between \(-5\) and \(2\) to indicate all numbers in this range are part of the interval.

Key Concepts

Interval Notation

Interval notation is a concise way of describing ranges of numbers. It eliminates the need for the verbose inequality symbols, making mathematical expressions more readable. In mathematics, intervals are used to describe the set of all numbers that lie between a pair of endpoints. When discussing interval notation, several key symbols are important to understand:

• The round bracket ( or parenthesis, e.g., \((-5, 2)\), is used to denote that an endpoint is not included in the interval.
• The square bracket \([ \) or \( ] \) indicates inclusion of that endpoint.

In this problem, we represent the inequality \(-5 < x < 2\) using the interval \((-5, 2)\).

• The absence of the number -5 and 2, which is shown by using round brackets, signifies that the endpoints are not part of the solution set.

Using interval notation helps simplify complex expressions and makes them more visually manageable.

Number Line Graph

A number line graph is a visual representation of an interval or set of numbers, helping to clearly display which values are included or excluded.
It offers an intuitive understanding of where the numbers fall in regards to a particular interval.
To graph the interval \((-5, 2)\), follow these straightforward steps:

• Draw a horizontal line and mark numbers at regular intervals along it to represent the number line.
• Identify the points -5 and 2 and place an open circle or dot at each of these numbers. This shows these endpoints are not included in the interval.
• Shade the region between the open circles to represent all numbers between -5 and 2. All values in this shaded area are included in our interval.

By using a number line, you can easily visualize the solution to an inequality, and it provides a clear distinction of included versus excluded numbers.

Open Interval

An open interval is a range of numbers where the endpoints are not included in the set of solutions. The concept of an open interval is vital for understanding inequalities where values are strictly less than or greater than a given number.
Open intervals are depicted using parentheses in interval notation, as seen in the expression \((-5, 2)\). Here, both -5 and 2 are excluded from the range of numbers that satisfy the inequality.

• Open circles are used in number line graphs to visually indicate that endpoints are not part of the set.
• Open intervals contrast with closed intervals, where brackets are used, indicating that the endpoints are included in the solution.

Understanding the difference between open and closed intervals is essential when working with intervals in mathematics. This understanding allows for the proper graphical and notation representation of inequalities.