Problem 13 - Precalculus: Mathematics for Calculus

Question

Graph the inequality. $$x<2$$

Step-by-Step Solution

Answer

Use an open circle at 2 and shade all points to the left on the number line.

Step 1: Understand the Inequality

An inequality is a mathematical sentence that shows the relationship between quantities that are not equivalent. In this case, we have the inequality \(x < 2\), which indicates all values that are less than 2.

Step 2: Identify the Boundary Point

The inequality \(x < 2\) has a boundary point at \(x = 2\). This is the point you will use as a reference when graphing.

Step 3: Choosing the Correct Symbol

Since the inequality is less than \(x < 2\) and does not include 2, we will use an open circle on the number line at \(x = 2\). Open circles indicate the value is not included.

Step 4: Determine the Shaded Region

Because the inequality is \(x < 2\), shade the region to the left of \(x = 2\) on the number line. This represents all values less than 2.

Step 5: Graph the Solution

Draw a number line, mark the point \(x = 2\) with an open circle, and shade the region to the left. This shows all the points on the number line that satisfy the inequality \(x < 2\).

Key Concepts

Graphing Inequalities

Graphing inequalities is a visual way to represent mathematical expressions that show the relationship between quantities that aren't equal. Let's take the inequality \(x < 2\) as an example. Here are the steps to graph an inequality on a number line:

• Identify the inequality symbol (such as <, >, ≤, or ≥) to understand the relationship. For \(x < 2\), it means the values of \(x\) are less than 2.
• Choose how to represent this on a number line. A number line is a straight horizontal line with numbers placed at intervals.
• Use an open circle or a closed dot based on whether the number is included in the inequality or not. For example, with \(x < 2\), we do not include 2, so we use an open circle at \(x = 2\).

With these basics, you can effectively translate any simple inequality into a graphical format using a number line.

Number Line Representation

A number line representation is a simple way to show the range of possible solutions for an inequality. When plotting these solutions for an inequality like \(x < 2\), follow these steps:First, draw a horizontal line and mark numbers at regular intervals on it. This creates a visual reference point.

• Locate the critical value related to the inequality, which is 2 in this case.
• Place an open circle directly above the 2 on this line. The open circle signifies that 2 itself is not part of the set of solutions.
• Shade the area or draw an arrow extending to the left of the open circle. This shaded region or arrow points to all the numbers that satisfy the condition \(x < 2\).

With these simple steps, the number line becomes a powerful tool to visually display inequality solutions.

Boundary Points

Boundary points are crucial when graphing inequalities because they define the edges of your solution set on a number line. In \(x < 2\), the boundary point is at \(x = 2\):

• A boundary point is where an inequality changes from true to false, or vice versa. It acts as a reference that shows the threshold for acceptable values.
• Determine whether to include the boundary point. Since \(x < 2\) excludes 2, we use an open circle to mark this key point. This open circle communicates that 2 itself is not part of any solution to \(x < 2\).
• Knowing where to place the boundary point helps ensure the inequality is represented accurately.

Understanding boundary points helps clarify why specific parts of the number line are shaded or left out, ensuring a correct and complete graph of an inequality.