Problem 1 - Precalculus: Mathematical for Calculus

Question

Graph the inequality. $$x<3$$

Step-by-Step Solution

Answer

Graph an open circle at 3 and shade left.

Step 1: Understanding the Inequality

The inequality given is \( x < 3 \). This means we want the values of \( x \) that are less than 3. In other words, \( x \) can be any number to the left of 3 on the number line.

Step 2: Sketching the Number Line

Draw a horizontal number line with marks at integer intervals. Be sure to include the point at 3 and several points to the left and right of it (such as 2, 1, and 4, 5).

Step 3: Locating the Boundary Point

Place a circle around the number 3 on the number line. Since the inequality is '<' (less than) and not '<=' (less than or equal to), we will use an open circle to indicate that 3 is not included in the solution.

Step 4: Shading the Solution Region

Shade the entire region to the left of the open circle around the 3. This represents all numbers that are less than 3, satisfying the inequality \( x < 3 \).

Key Concepts

Number Line

A number line is a straight horizontal line that allows you to visually represent numbers. Every point on the line corresponds to a unique real number.
For graphing simple inequalities like the one we have, it’s important to include several points to clearly show the relevant range.

• Draw marks at regular intervals, typically at integer values.
• Label at least the number directly involved in the inequality, and a few numbers on either side.

This setup helps to immediately see which numbers are part of the solution. In our scenario, we would draw a number line and clearly mark numbers such as 2, 3, 4, and so forth. This visual aid will aid understanding by providing context and structure to the inequality.

Solving Inequalities

Solving inequalities involves finding the set of all real numbers that satisfy the inequality. In our case, we are looking at the inequality \( x < 3 \).

• Identify the inequality sign. A less than '<' sign means numbers less than a certain value.
• The inequality \( x < 3 \) implies that \( x \) includes all numbers to the left of 3 on the number line.

Solving the inequality doesn't always provide a specific value but identifies a range of values, or the interval, where the solution lies. Setting up this inequality visually on a number line can be useful for understanding the range of potential solutions.

Open and Closed Circles

Graphing inequalities often involves using open and closed circles to indicate whether endpoints are included in the solution set.

• An open circle is used for inequalities using '<' or '>', showing that the number is not a part of the solution set.
• A closed circle is reserved for inequalities using '≤' or '≥', indicating the number is included in the solution.

In the inequality \( x < 3 \), we use an open circle at 3, signifying that 3 itself is not a solution. The shading to the left shows all numbers less than 3 that satisfy this inequality. Open and closed circles thus help in clearly indicating the range of solutions and refining which numbers it includes or excludes.