Problem 49 - Algebra and Trigonometry

Question

\(47-52\) : Express the inequality in interval notation, and then graph the corresponding interval. $$ -2

Step-by-Step Solution

Answer

The interval is \((-2, 1]\).

Step 1: Analyze the Inequality

The given inequality is \(-2 < x \leq 1\). This indicates that \(x\) is greater than \(-2\) and less than or equal to \(1\).

Step 2: Determine Interval Format

Based on the inequality \(-2 < x \leq 1\), use a parenthesis for \(-2\) since it is not included (\(<\) symbol) and a bracket for \(1\) since \(x\) can equal to \(1\) (\(\leq\) symbol).

Step 3: Express in Interval Notation

The interval is \((-2, 1]\). This means all numbers between \(-2\) and \(1\), including \(1\) but not \(-2\), are part of the interval.

Step 4: Graph the Interval

Draw a number line and represent the interval \((-2, 1]\) by marking an open circle at \(-2\) (to show \(-2\) is not included) and a closed dot at \(1\) (to show \(1\) is included). Then shade the region between them.

Key Concepts

Interval Notation

Interval notation is a way of representing the set of numbers that satisfy a particular range or inequality. It uses parentheses and brackets to show whether endpoints are included or excluded.
In interval notation,

• Parentheses ( ) are used when the number is not included.
• Brackets [ ] are used when the number is included.

For the inequality \(-2 < x \leq 1\), the interval notation is written as \((-2, 1]\). Here, we use a parenthesis around \(-2\) because \(-2\) is not part of the solution (\(<\) symbol), and a bracket around \(1\) because \(x\) can equal \(1\) (\(\leq\) symbol).
Interval notation offers a concise way to convey ranges and can be helpful in understanding the scope of solutions to a problem.

Graphing Inequalities

Graphing inequalities helps visualize the range of values that satisfy the inequality. You represent inequalities on a number line as intervals.
The method involves the following steps:

• Identify the inequality signs involved.
• Use open circles for strict inequalities (< or >), meaning the endpoint is not included.
• Use closed circles (or solid dots) for inclusive inequalities (\leq or \geq), meaning the endpoint is included.

Once these circles or dots are placed, fill in, or shade, the part of the number line between the endpoints to illustrate all numbers included in the solution.
For example, for the interval \((-2, 1]\), we place an open circle at \(-2\) and a closed circle at \(1\), then shade the number line in between to indicate all numbers \(x\) that satisfy the condition \(-2 < x \leq 1\).
This visual representation supports a clear understanding of the inequality's reach.

Number Line Representation

A number line provides a simple, visual way to show the range of an inequality. It clearly marks the inclusion or exclusion of boundary values and the solution set.
Here's how to use a number line for the inequality \(-2 < x \leq 1\):

• Draw a horizontal line to represent the number line.
• Plot key points \(-2\) and \(1\) on this line.
• Place an open circle at \(-2\) to show that \(-2\) is not part of the solution.
• Use a closed circle at \(1\) because \(1\) is included in the solution (inclusivity).
• Shade the portion of the line located between \(-2\) and \(1\) to demonstrate all values that \(x\) can take.

This method makes it easier for students and anyone tackling inequalities to visually assess the range and limitations of the solution set.