Problem 9 - Elementary Algebra

Question

Graph all solutions on a number line and provide the corresponding interval notation. $$ x<-12 $$

Step-by-Step Solution

Answer

The solutions are on the interval \((-\infty, -12)\).

Step 1: Understanding the Inequality

The inequality is given as \(x < -12\). This means that we need to find all numbers less than \(-12\).

Step 2: Representing the Solution on a Number Line

To graph the solution \(x < -12\), we draw a number line, and place an open circle at \(-12\) to indicate that \(-12\) is not included. Then, shade all the numbers to the left of \(-12\). This visually represents all numbers less than \(-12\).

Step 3: Writing the Interval Notation

In interval notation, \(x < -12\) is represented by the interval \((-\infty, -12)\). The parenthesis '(' is used to show that \(-12\) is not included, and \(-\infty\) always has a parenthesis because infinity is never included.

Key Concepts

Number Line

The number line is an essential tool in understanding and visualizing inequalities. It is a straight, horizontal line where each point represents a real number. Zero is typically in the center, with positive numbers to the right and negative numbers to the left. When dealing with inequalities like \(x < -12\), we use the number line to show all the possible solutions.
On the number line, you place notable points based on the context of the problem. In the case of \(x < -12\), you identify the position of \(-12\). To show the solutions for the inequality, you use an open circle at \(-12\). An open circle means that \(-12\) itself is not a solution, but all numbers less than \(-12\) are solutions. From there, you shade or darken the portion of the line to the left of \(-12\) to indicate all these numbers are included in the solution set.
This visual method helps in understanding which numbers satisfy the conditions given by an inequality.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It's a compact and readable way to express the range of numbers that work in a given inequality. It's especially helpful for communicating which endpoints are included or excluded.
For our inequality \(x < -12\), the interval notation is \((-\infty, -12)\). Here's how it works:

• The use of parentheses indicates that the endpoint, \(-12\), is not included in the solution set. Parentheses mean the set is open at this point.
• The symbol \(-\infty\) represents all numbers going infinitely to the left on the number line.
• Infinity symbols (both positive and negative) are always accompanied by parentheses, as infinity itself is a concept, not a number that can be reached or included.

This notation style clearly communicates that every number from negative infinity up to, but not including, \(-12\) is part of the solution set to the inequality.

Open Circle

An open circle on the number line is used to highlight that a particular number is not part of the solution set in an inequality. It serves as a visual indicator of exclusion for students and anyone interpreting the graph.
When you see \(x < -12\), it tells you that \(-12\) is the boundary but is not part of the solution. To indicate this graphically, an open circle is used at \(-12\). The open nature of the circle shows clearly that \(-12\) itself doesn't satisfy \(x < -12\).
In contrast, if the inequality was inclusive like \(x \leq -12\), a closed circle would be employed to show that \(-12\) itself is included. Thus, understanding the difference between open and closed circles is key when analyzing and graphing inequalities.