Problem 37 - Algebra and Trigonometry

Question

\(33-38\) . Express the interval in terms of inequalities, and then graph the interval. $$ [2, \infty) $$

Step-by-Step Solution

Answer

The interval \([2, \infty)\) is expressed as the inequality \(x \geq 2\), and is graphically represented by a solid dot at 2 with an arrow extending to the right.

Step 1: Understand the Interval Notation

The interval notation \([2, \infty)\) includes all real numbers starting from 2 and extending infinitely to the right. The square bracket "[" indicates that 2 is included in the interval, while the parenthesis ")" indicates that infinity is not a fixed number and thus cannot be included.

Step 2: Express the Interval as an Inequality

To convert the interval notation \([2, \infty)\) into inequality form, recognize that this means all numbers, \(x\), that are greater than or equal to 2. This can be expressed as the inequality: \(x \geq 2\).

Step 3: Graph the Inequality

To graph the inequality \(x \geq 2\), draw a number line. Place a solid dot at \(x = 2\) to represent that 2 is included in the interval. Then, draw an arrow extending to the right from 2, representing all values greater than or equal to 2.

Key Concepts

Inequality

An inequality is a mathematical statement that compares two values. It shows the relationship between them by using symbols like \( < \), \( > \), \( \leq \), or \( \geq \). These symbols help us understand whether the values are less than, greater than, or equal to each other.
For instance, the inequality \( x \geq 2 \) means that \( x \) is greater than or equal to 2. This implies that 2 is the smallest value that \( x \) can take, but it can also take any value greater than 2.
Inequalities are helpful in various mathematical contexts because they provide a way of expressing ranges of possible values, which is often used in solving real-world problems, from economics to engineering.

Number Line

A number line is a visual tool used in mathematics to represent numbers in a linear fashion. It's essentially a straight line with numbers placed at equal intervals along it. Each point on the line corresponds to a real number; this helps with visualizing math concepts, like inequalities.
To plot an inequality like \( x \geq 2 \) on a number line:

• First, identify the point for 2 on the line.
• Use a solid dot at 2 to show that it's included in the solution set based on inequality rules.
• Then draw an arrow going to the right, showing all numbers greater than 2 are part of the solution.

The number line gives a clear picture of where the values lie in relation to each other. It makes it easier to see which numbers are solutions to the given inequality.

Graphing Inequalities

Graphing inequalities is about showing the solution set of an inequality on a number line. It turns the abstract mathematical statements into something you can see and understand visually.
Here's how to graph \( x \geq 2 \):

• Draw a horizontal line representing the number line.
• Locate the number 2 and place a solid dot since the inequality includes \( x = 2 \) (i.e., it's \( \geq \), not just \( > \)).
• Add an arrow extending to the right from the dot, indicating all numbers greater than 2 are included.

This graphical representation is significant as it helps in quickly ascertaining the set of possible solutions visually. Graphs of inequalities are particularly useful in higher mathematics where multiple inequalities might be dealt with simultaneously.

Real Numbers

Real numbers include all the numbers on the number line. This encompasses whole numbers, fractions, decimals, and irrational numbers. Essentially, any point on an infinite line is a real number.
The interval \([2, \infty)\) refers only to real numbers starting from 2 and moving rightward. These are numbers we commonly use in everyday arithmetic, such as 2, 3.5, or \(\pi\).
Understanding real numbers is fundamental, as they form the basic set of numbers used in most mathematical operations and concepts. They enable more advanced maths by allowing concepts like continuity, limits, derivatives, and integrals. This broad set of numbers provides a comprehensive framework necessary for both theoretical and practical applications.