Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph the solutions of each inequality on a number line. $$x<-2$$

Step-by-Step Solution

Answer

The solutions of the inequality \(x < -2\) are represented on the number line with an open circle at -2 and a line or arrow pointing to the left from -2, indicating all numbers less than -2.

Step 1: Understand the inequality

First, let's understand what the inequality \(x < -2\) means. It means that we are looking for all values of \(x\) that are less than -2.

Step 2: Draw a number line

Next, draw a straight horizontal line, which will represent our number line. Mark the point -2 on the line.

Step 3: Represent the inequality on the number line

Since \(x\) is less than -2 and not equal to -2, we put an open circle on -2. An open circle means that -2 is not included in the solution set. Then draw a line or arrow pointing to the left from -2, to indicate all numbers less than -2.

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities, the goal is to visually represent a range of values that satisfy the given inequality condition. In our example, we have the inequality \(x < -2\). This asks us to find every possible value of \(x\) that is less than -2.

To graph this on the number line, you will need to:

• Draw a horizontal line, which we call the number line.
• Locate and mark the point -2 on this line.
• Use an open circle to indicate that -2 itself is not part of the solution.
• Draw a line extending left from the open circle to represent numbers less than -2.

This approach helps visually understand which values satisfy \(x < -2\), giving a clear graphical representation of the solution set.

Understanding Inequalities

Inequalities describe relationships between expressions that aren't exactly equal. They use symbols like \(<, >, \leq, \) and \(\geq\). Let's focus on \(<\) for our example.

The inequality \(x < -2\) suggests that \(x\) can be any number smaller than -2. It covers an infinite range of values and isn't restricted to a single number.

Here's a quick breakdown of inequality symbols:

• "\(<\)" means less than. \(x < y\) implies that \(x\) is smaller than \(y\).
• "\(>\)" means greater than. \(x > y\) means \(x\) is larger than \(y\).
• "\(\leq\)" means less than or equal to. Here the boundary value is included.
• "\(\geq\)" means greater than or equal to. The boundary value is part of the solution.

Inequalities show flexible relationships where solutions can be many different values within a certain range.

Number Line Representation

A number line is a simple yet powerful tool for visualizing inequalities. It makes it easier to understand which values satisfy the inequality, especially when dealing with variables.

In our case, on the number line:

• -2 is marked with an open circle, signifying \(x < -2\).
• The open circle indicates that -2 isn't included in the set of solutions, as opposed to a closed circle which would indicate inclusion.
• An arrow or a line to the left of the open circle captures the essence of values continuously decreasing beyond -2.

Whether in homework or more complex equations, the number line is your ally in pinpointing which regions of numbers meet an inequality's conditions. By learning to use this representation, you strengthen your understanding of mathematical relationships.