Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ -2

Step-by-Step Solution

Answer

The interval is \((-2, 4]\). Open circle at \(-2\), closed circle at \(4\), shade between.

Step 1: Understand the Inequality

The inequality given is \(-2 < x \leq 4\). This tells us that \(x\) is greater than \(-2\) and at the same time \(x\) is less than or equal to \(4\).

Step 2: Express the Inequality in Interval Notation

To express \(-2 < x \leq 4\) in interval notation, we use a parenthesis to indicate that \(-2\) is not included and a bracket to show that \(4\) is included. The interval is written as \((-2, 4]\).

Step 3: Sketch the Graph of the Interval

To sketch \((-2, 4]\), draw a number line. Place an open circle at \(-2\) to show it is not included and a closed circle at \(4\) to indicate it is included. Shade the region between \(-2\) and \(4\) to represent all values \(x\) can take.

Key Concepts

Interval Notation

Interval notation is a shorthand method for representing inequalities in algebra. It provides a concise way to express ranges of values that a variable can take. In our exercise, we have the inequality \(-2 < x \leq 4\), which translates into interval notation as \((-2, 4]\). Here’s how it works:

• A parenthesis ( indicates that the endpoint is not included in the interval. This is known as an open interval.
• A bracket [ signifies that the endpoint is included. This is called a closed interval.

In the case of \(-2 < x \leq 4\), we use a parenthesis for \(-2\) because \(-2\) itself is not a possible value for \(x\). Conversely, since \(4\) is part of the acceptable values, we use a bracket for it. Interval notation is both efficient and helpful when you have to write the domains of functions or solutions to inequalities.

Graphing Inequalities

Graphing inequalities involves representing the solution set of an inequality on a number line. This visual approach helps in understanding which values of the variable satisfy the inequality condition. Let’s break down the process using our inequality example \(-2 < x \leq 4\):

• First, draw a horizontal line to represent the number line.
• Identify and mark the critical points, which are \(-2\) and \(4\) in our example.
• Use an open circle at \(-2\) to show that \(-2\) is not included in the solution set.
• Place a closed circle at \(4\) to indicate that it is included.

After marking these points, shade the region between \(-2\) and \(4\) to depict all possible values of \(x\) that make the inequality true. Graphing provides a clear visual confirmation of the interval, facilitating a better grasp of the range of solutions.

Number Line Representation

The number line is an essential tool in algebra for visually representing intervals and inequalities. Here’s a step-by-step way to use this tool effectively:

• Begin by drawing a horizontal line, labeling it with regular intervals of numbers.
• Locate your critical points on this line. For \(-2 < x \leq 4\), these points are \(-2\) and \(4\).
• Use symbols such as circles to indicate whether an endpoint is included in the interval. Open circles mean the number isn't part of the set, while closed circles indicate inclusion.
• Shade or highlight the section of the number line fitting within your inequality. This shaded area visually represents all values that satisfy the inequality.

By using simple lines and symbols, a number line offers a clear, intuitive way to understand and communicate inequalities. It simplifies complex concepts by allowing visual representation, which aids in comprehension.