Problem 20 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$(3, \infty)$$

Step-by-Step Solution

Answer

The inequality is \(x > 3\). This is represented on a number line as an open circle at 3, with the line shaded to the right of 3.

Step 1: Convert to Inequality

The interval notation (3, ∞), can be expressed as an inequality by keeping on mind that the parentheses mean the number 3 is not included. This can be written as the inequality: \( x > 3 \). This means 'x is greater than 3'.

Step 2: Graph on a Number Line

Start by drawing a simple number line, and marking 3. Because the inequality is 'greater than' but not 'equal to', make an open circle at 3. Then, because the inequality indicates that x is greater than 3, shade the number line to the right of 3.

Key Concepts

Understanding Inequalities

An inequality is much like an equation but instead of an equal sign, it uses inequality symbols like \(<\), \(>\), \(\leq\), or \(\geq\). These symbols show that the two sides of the inequality are not exactly equal. For example, \(x > 3\) means 'x' can be any number greater than 3.

Unlike equations, inequalities offer a range of possible solutions rather than a single solution. This is important to understand as it allows you to express sets of numbers that satisfy a specific condition.

• \(<\) and \(>\) mean the variable can be less or more than a specific number, but not equal to it.
• \(\leq\) and \(\geq\) mean the variable can also be equal to the specified number, providing a bit more flexibility.

Grasping this notion is crucial when dealing with a wide range of mathematical problems.

Using a Number Line

A number line is a simple and effective tool for visualizing numbers and inequalities. It helps you see the relationships between different numbers quickly.

To represent an inequality on a number line, you need to know whether to use an open or closed circle. Here’s how to go about it:

• An open circle is used for \(<\) or \(>\) inequalities, showing that a number is not included in the solution.
• A closed circle is for \(\leq\) or \(\geq\), indicating the number itself is part of the solution.

Position the number on the line according to its value, ensuring clarity and precision in your visualization. This makes it easier to grasp the range of solutions an inequality represents.

Graphing Inequalities

Graphing inequalities is about putting your findings onto a number line to represent the solution set visually. It's a helpful process for understanding which numbers fulfill the inequality.

Using the previous example of \(x > 3\), we'd start our graph at the number 3. Here's how you can do it step-by-step:

• Draw a horizontal line and mark key numbers, in this case, the number 3.
• Since \(x > 3\) means 'greater than', place an open circle at 3 to show that 3 is not part of the solution set.
• Shade the line to the right of the 3. This indicates that all numbers greater than 3 satisfy the inequality.

This visual representation allows you to see the infinite possibilities that fit the inequality, making it a crucial skill in mathematics.