Problem 5 - Algebra for College Students

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x<-2\)

Step-by-Step Solution

Answer

The inequality is \((-\infty, -2)\); graph: open circle at \(-2\) and shade left.

Step 1: Understand the Inequality

The given inequality is \(x < -2\), which means that \(x\) includes all real numbers that are less than \(-2\). It does not include \(-2\) itself.

Step 2: Express the Inequality in Interval Notation

To express the inequality \(x < -2\) in interval notation, we need to identify the starting and ending points of the interval. Since \(x\) can be any number less than \(-2\), the interval starts from \(-\infty\) to \(-2\). However, \(-2\) is not included in the interval. In interval notation, this is written as \((-\infty, -2)\).

Step 3: Sketch the Graph of the Interval

To sketch the graph of the interval \((-\infty, -2)\), draw a number line, place a point at \(-2\), and shade the line to the left of \(-2\) to indicate all numbers less than \(-2\). Use an open circle at \(-2\) because \(-2\) is not included in the interval.

Key Concepts

Interval Notation

Interval notation is a concise way of representing inequalities or ranges of numbers on a number line. It captures information about which numbers are included or excluded from an interval.
When we express an inequality like \(x < -2\) in interval notation, we are looking for all possible values that satisfy this inequality. In this case, \(x\) includes all numbers less than \(-2\).

Here's how interval notation works:

• Parentheses \((\ , \ )\) indicate that the endpoint is not included. This is called open interval.
• Brackets \([\ , \ ]\) indicate that the endpoint is included. This is a closed interval.
• An interval can also be a mix of parentheses and brackets if one end is included and the other is not.
• The symbol \(-\infty\) is always paired with a parenthesis because infinity is not a specific number; it's a concept indicating boundlessness and can't be included, thus always used as \(( -\infty, a )\) or \(( a, \infty )\).

For \(x < -2\), the interval notation is \(( -\infty, -2 )\), showing \(-2\) is not included.

Number Line

A number line is a straight line where each point corresponds to a real number. It is an essential tool for visualizing mathematical concepts, such as inequalities and intervals.
To understand \(x < -2\) on a number line, imagine a straight horizontal line where 0 is usually placed in the center. Negative numbers (-1, -2, -3, ...) extend to the left, and positive numbers (1, 2, 3, ...) extend to the right.

Construction of a number line includes:

• Choosing an appropriate scale to mark numbers in sequence.
• Placing negative numbers to the left of zero and positive numbers to the right.
• Using circles to indicate whether a number is included in the interval (a filled circle for inclusion, open circle for exclusion).

For the inequality \(x < -2\), we place an open circle at \(-2\) to show it's not included, and shade the line to the left of \(-2\) indicating all numbers less than \(-2\) are part of the solution.

Graph of Intervals

Graphing intervals on a number line provides a clear visual method for understanding the extent and limitation of a set of numbers. It is especially useful for comprehending which values satisfy a given inequality.
The graph of intervals makes use of a number line, where one highlights or shades regions that represent the interval.

When graphing the interval \( (-\infty, -2) \), follow these steps:

• Start by drawing a horizontal line to represent the number line.
• Locate \(-2\) on the line.
• Place an open circle at \(-2\) to show it is not included in the interval.
• Shade the area towards the left, away from \(-2\), indicating that the interval comprises all numbers less than \(-2\).

This shaded region visually displays all the solutions to the inequality \(x < -2\), making it easier to understand which values are part of the interval.