Problem 7 - Precalculus with Limits

Question

In Exercises 5-18, sketch the graph of the inequality. $$x \geq 6$$

Step-by-Step Solution

Answer

The graph of the inequality \(x \geq 6\) is a number line with a solid line on 6 and everything to the right of that line shaded or highlighted.

Step 1: Understand the inequality

The inequality \(x \geq 6\) means that all x-values greater than or equal to 6 will satisfy the inequality. This forms a semi infinite region, from 6 to positive infinity, including 6.

Step 2: Sketch the graph

Start by sketching a number line. The number line will have 6 and all numbers greater than 6 highlighted because these are the x-values that satisfy the inequality. Because the inequality is \(x \geq 6\), we also include the number 6, which is represented by a solid dot or line on 6.

Step 3: Finalize the graph

You should now have a number line with a solid line on 6 and everything to the right of that line shaded or highlighted, representing all numbers greater than or equal to 6.

Key Concepts

number line

When learning to graph inequalities, the number line plays a central role. It is a simple tool that helps visualize which numbers satisfy a given inequality.
To graph an inequality like \(x \geq 6\), one must first draw a horizontal line, which represents the number line. This number line typically includes evenly spaced marks to represent integers.

• Start by marking the point 6, as it is the critical value in the inequality \(x \geq 6\).
• Include a wide enough range to show all relevant numbers greater than this point.

A number line is particularly useful in showing continuous data and can help students understand the infinite nature of inequalities that extend beyond this line, like moving towards positive infinity in this case.

inequality symbols

Understanding inequality symbols is essential for correctly graphing inequalities. In this case, the inequality \(x \geq 6\) involves two parts to understand: the variable \(x\), and the inequality symbol itself.
The symbol \(\geq\) is read as "greater than or equal to". This means that \(x\) can be any number that is either more than or exactly equal to 6.

• The \(>\) part stands for "greater than", which allows values larger than 6.
• The \(=\) part is crucial because it includes the boundary point (6) itself in the solution set.

Familiarity with inequality symbols is crucial to correctly interpret and notate mathematical statements. Different symbols denote different sets of numbers that solve the inequality – for example, \(<\) means less than, and \(\leq\) means less than or equal to.

shading regions

Shading is an effective way to visually represent the range of numbers that satisfy an inequality on a number line. When graphing \(x \geq 6\), shading indicates all possible solutions that \(x\) can take.
Here’s how you shade the correct region for \(x \geq 6\):

• First, draw a solid dot at the number 6 to show that 6 is included in the solution set. This indicates the "equal to" part of \(\geq\).
• Next, shade or highlight the entire number line extending to the right of 6. This shaded portion represents all numbers greater than 6.

Always remember that shading is not just a finishing touch – it’s a crucial visual element that provides clarity about what numbers are part of the solution. For different inequalities, the shading might extend in the opposite direction or may not include a boundary point, but this approach will help you get started.