Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ x<-2 $$

Step-by-Step Solution

Answer

The interval is \((-\infty, -2)\), and it is graphed with an open circle at -2, shading left.

Step 1: Understand the Inequality

The given inequality is \( x < -2 \). This means that \( x \) represents any number that is less than \(-2\). Numbers less than \(-2\) are all those to the left of \(-2\) on the number line.

Step 2: Write the Inequality as an Interval

In interval notation, an inequality \( x < a \) is expressed as \((-\infty, a)\). Since our inequality is \( x < -2 \), we write it as the interval \((-\infty, -2)\). The round bracket \((\) indicates that \(-2\) is not included in the set of solutions.

Step 3: Sketch the Graph

To sketch the graph of \( x < -2 \), draw a number line. Highlight or shade the section to the left of \(-2\) to indicate all numbers less than \(-2\). Place an open circle at \(-2\) to show that \(-2\) is not included in the solution set.

Key Concepts

Interval Notation

Interval notation is a method of writing down a set of numbers that represent all possible solutions to an inequality. It is a concise way to express the range of values a variable can take. For example, in the inequality \( x < -2 \), \( x \) is any number less than \(-2\). To express this in interval notation, we write \((-\infty, -2)\). This notation tells us two things:

• The range begins at negative infinity (\(-\infty\)), meaning there is no lower limit to how small the numbers can be.
• The number \(-2\) is not included in the range, indicated by the use of a round bracket \(()\) rather than a square bracket \([]\).

Interval notation is useful because it provides a clear, unambiguous way to communicate inequalities.

Number Line

A number line is a visual tool used in mathematics to represent numbers in a linear format. It is especially useful for illustrating solutions to inequalities. On a number line, each point corresponds to a real number, and these are arranged in order from left to right. For example, to represent the inequality \( x < -2 \) on a number line:
- Start by locating the point that represents \(-2\). - Since \( x \) is less than \(-2\), identify all points to the left of \(-2\) as part of the solution.
A number line simplifies understanding of the relationship between numbers and makes it easy to see which regions represent solutions to inequalities.

Graphing Inequalities

Graphing inequalities involves showing the solution of an inequality on a number line or coordinate plane. For a simple inequality like \( x < -2 \), this process is straightforward:

• Draw a number line and mark the point \(-2\).
• Shade the region to the left of \(-2\) to indicate all numbers less than \(-2\) are solutions.

This shaded area visually represents where the inequality is true. By graphing inequalities, it's easier for students to visualize the range of possible solutions and understand how inequalities can be conveyed in a two-dimensional space.

Open Circle

An open circle on a number line is used to indicate that a specific number is not included in the solution set of an inequality. For the inequality \( x < -2 \):

• Place an open circle at the point \(-2\) on the number line.
• This visually signals that although \(-2\) is the boundary of the solution set, it is not part of the set.

The open circle is an important detail when graphing inequalities, as it distinguishes between "less than" or "greater than" (exclusive) and "less than or equal to" or "greater than or equal to" (inclusive). By using open circles, we clearly communicate the nature of the boundary in our solution.