Problem 4 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

In interval notation: \([3, \infty)\).

Step 1: Understand the Inequality

The inequality given is \(x \geq 3\). This means that \(x\) can take any value that is 3 or greater than 3.

Step 2: Convert to Interval Notation

The inequality \(x \geq 3\) includes all numbers from 3 to infinity. In interval notation, this is expressed as \([3, \infty)\). The square bracket [ indicates that 3 is included in the interval, while the parenthesis ) indicates that infinity is not included.

Step 3: Sketch the Graph

To sketch the graph of \([3, \infty)\), draw a number line. Place a closed dot or solid circle at 3 to show that 3 is included. Then, draw a line or arrow extending to the right towards infinity, indicating that all numbers greater than 3 are part of the interval.

Key Concepts

Inequality

An inequality is a mathematical statement that indicates the relationship between two values, showing that one value is larger or smaller than the other. In inequalities, we use symbols such as \( <, >, \leq, \) and \( \geq \). The inequality \( x \geq 3 \) is a type of inequality called "greater than or equal to." It tells us that the variable \( x \) can be either exactly 3 or any number larger than 3. This kind of inequality expresses a range of possible values for \( x \), not just a single solution as equations do.

When you interpret inequalities, think of them as conditions or rules that numbers must follow, providing flexibility in solutions. Understanding inequalities is essential as they are used in various fields, such as in budgets, constraints, and many optimization problems.

Graphing Inequalities

Graphing inequalities is a visual way to represent all the possible solutions of an inequality on a number line. For the inequality \( x \geq 3 \), we want to show every number that satisfies this condition. To perform this successfully, we begin by marking the specific point, in this case, 3, on a number line. Since the inequality is "greater than or equal to," we depict this by using a closed dot or a solid circle at 3.

After marking 3, draw an arrow extending to the right of the number line. This arrow represents all the numbers greater than 3, stretching towards positive infinity. By illustrating inequalities on a number line, you can quickly and clearly show which numbers meet the inequality condition. Graphs make it easier to see the solution set visually, especially when dealing with more complex inequalities or systems of inequalities.

Number Line

A number line is a straight horizontal line that is a visual representation of numbers placed at regular intervals. It allows us to compare and plot numbers easily. Each point on a number line corresponds to a real number.

When working with inequalities like \( x \geq 3 \), the number line becomes a helpful tool to express the solution set. To create a number line, write equal spaced numbers in the positive and negative directions. You start at a central point, usually 0, and extend it to include any numbers needed for the inequality problem you are tackling.

• Closed or open points on the number line demonstrate whether a boundary value is included in the set (closed for inclusion, open for exclusion).
• Arrows show continuous ranges and where the inequality is valid.

Utilizing a number line simplifies the process of understanding and solving inequalities and is fundamental in learning basic algebra.