Problem 5 - Algebra A Combined Function

Question

Graph each inequality on the number line. $$ y<4 $$

Step-by-Step Solution

Answer

Shade the number line to the left of an open circle at 4.

Step 1: Understand the Inequality

The inequality given is \( y < 4 \). This means that \( y \) represents all numbers that are less than 4.

Step 2: Identify the Position on the Number Line

On the number line, the number 4 is a point where \( y \) must be less than, not equal to. Thus, 4 is a boundary point.

Step 3: Use an Open Circle

Since \( y \) is less than 4 and not equal to it, represent the boundary point, 4, with an open circle to indicate that 4 is not included in the solution set.

Step 4: Shade the Correct Side of the Number Line

Shade the number line to the left of 4. This shading represents all the values that are less than 4.

Key Concepts

number line

In mathematics, a number line is a visual representation of numbers along a horizontal line. It helps us understand the relative position and magnitude of numbers. When solving inequalities like \( y < 4 \), the number line becomes essential.Here's how it works:

• The number line is divided into equal segments that correspond to increments of one unit, making it easier to identify points like 0, 1, 2, 3, and so on.
• It extends in both directions infinitely, with negative numbers extending to the left of zero and positive numbers to the right.
• In the context of inequalities, the number line helps us visually identify all values that satisfy the inequality.

To graph the inequality \( y < 4 \) on the number line, you would first locate the number 4 as a reference point. Then, use the concepts of open circle and shading to visually represent the solution set.

open circle

An open circle is a critical symbol used on the number line when graphing inequalities. It helps us communicate the idea of boundary points that are not included in the solution. For the inequality \( y < 4 \), an open circle is used at the number 4.Here’s why an open circle is important:

• It indicates that the number at this point (in this case, 4) is not part of the solution set that satisfies the inequality. This means that while values like 3.9, 3.99, and smaller are part of the solution, the number 4 is not.
• An open circle ensures clarity, so there is no confusion about whether the boundary number is included.

Using an open circle tells us precisely where the solution starts (or stops) on the number line, and it is a fundamental part of graphing inequalities accurately.

solution set

In solving inequalities, the solution set is all the possible values that satisfy the inequality. For \( y < 4 \), the solution set includes every number that is less than 4.Here's how to determine the solution set:

• Identify if the inequality is strict (e.g., \( < \) or \( > \)) or includes equality (e.g., \( \leq \) or \( \geq \)). A strict inequality like \( y < 4 \) means 4 itself is not part of the solution, hence using an open circle at 4.
• The shaded part of the number line represents this solution set. For \( y < 4 \), we shade all numbers to the left of the open circle at 4 to denote the solution set.
• Examples of numbers in the solution set for this inequality include any value below 4, such as 3, 2, 0, -1, and so forth.

By representing inequalities and solution sets visually with a number line, open circles, and shading, we make the understanding and solving of inequalities more intuitive.