Problem 4 - Intermediate Algebra

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x \geq 3\)

Step-by-Step Solution

Answer

The inequality \(x \geq 3\) in interval notation is \([3, \infty)\).

Step 1: Understand the Inequality

The inequality given is \(x \geq 3\). This means that \(x\) is greater than or equal to 3. In this case, the set of all possible values for \(x\) includes 3 and all numbers greater than 3.

Step 2: Express in Interval Notation

To express \(x \geq 3\) in interval notation, we need to identify the lower and upper bounds. Since \(x\) includes 3 and continues indefinitely to the right, the interval is from 3 to infinity. Therefore, in interval notation, it is \([3, \infty)\). The square bracket [ indicates that 3 is included, and the parenthesis ) indicates that infinity is not included.

Step 3: Sketch the Graph

To sketch the graph of the interval \([3, \infty)\), draw a number line. Place a closed circle or a solid dot at 3 to indicate that 3 is included in the interval. Draw a line extending to the right towards infinity to indicate that all numbers greater than 3 are included. The direction and extent of this line show the set of all possible values of \(x\) for the inequality.

Key Concepts

Interval Notation

When we talk about interval notation, we're referring to a compact way of describing a range of values. It's especially useful for inequalities, as it simplifies how we express sets of numbers. Let's use the inequality \(x \geq 3\) as an example.

In this context:

• The interval starts at 3, because \(x\) can be 3 (due to the 'greater than or equal to').
• It extends to positive infinity, meaning there is no upper limit defined for \(x\).

This is written in interval notation as \([3, \infty)\).

• The square bracket \([\) next to 3 signifies that 3 is included in the set of values.
• The parenthesis \()\) next to infinity indicates infinity is never actually reached, so it's not included.

This notation is concise and makes it easy to see the start and end of the interval at a glance.

Number Line

Number lines are a visual way to represent numerical values. They help in understanding and interpreting inequalities like \(x \geq 3\). Drawing a number line gives you a perspective similar to a graph, with every point representing a number.

Here's how to use it for \(x \geq 3\):

• Draw a straight horizontal line. This represents all possible real numbers.
• Mark a point at 3 on the line—this is a reference for where your interval begins.
• Use a solid dot at 3 to show that the number 3 is included in this interval.
• From 3, draw an arrow or a continuous line extending to the right. This indicates that all numbers greater than 3 are included.

A number line aids in visualizing solutions to inequalities and gives a tangible representation of intervals, helping to easily grasp the concept.

Inequality Graphing

Inequality graphing is vital for visualizing solutions of inequalities like \(x \geq 3\). By graphing, we turn abstract mathematical concepts into concrete visual forms. This process involves a few straightforward steps that make understanding inequalities clearer.
For \(x \geq 3\):

• Begin with a number line, similar to what we discussed in the previous section.
• Identify the point (in this case, 3) where the inequality starts.
• Use a closed circle at 3 to signify inclusion in the interval.
• Extend a line or arrow to the right of 3, illustrating that all numbers greater than 3 satisfy the inequality.

The direction of the line or arrow represents the set of solutions that satisfy the inequality. This method of graphing inequalities helps students recognize and interpret solution sets effectively.