Problem 41 - Algebra and Trigonometry

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$ -2

Step-by-Step Solution

Answer

Interval notation: \((-2, 1]\). Graph: Shade between an open circle at -2 and a closed circle at 1.

Step 1: Understand the inequality

The inequality given is \(-2 < x \leq 1\). It consists of two parts: \(-2 < x\) and \(x \leq 1\). This tells us that \(x\) is greater than \(-2\) and less than or equal to \(1\).

Step 2: Express the inequality in interval notation

Interval notation represents the set of solutions to an inequality. For \(-2 < x\), \(-2\) is not included, represented by a parenthesis \((-\), and for \(x \leq 1\), \(1\) is included, represented by a bracket \(]\). In interval notation, the solution is \((-2, 1]\).

Step 3: Graph the interval

To graph \((-2, 1]\), draw a number line, place an open circle at \(-2\), indicating it is not included, and a closed circle at \(1\), indicating it is included. Shade the region between them to represent all numbers in that interval.

Key Concepts

Inequality

Inequalities are mathematical expressions that describe the relationship between two values. In our problem, the inequality given is \(-2 < x \leq 1\). This means that the value of \(x\) is greater than \(-2\) but less than or equal to \(1\).
There are various inequality symbols:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

These symbols help us to specify a range of values for an unknown variable. Inequalities are like guidelines—they tell us where within a range our solutions can be found. Understanding the specific requirements of each inequality symbol is crucial for translating these into interval notation and other forms of representation.
Knowing this, you can easily interpret similar mathematical situations.

Number Line Graph

A number line graph is a simple and clear way to visually represent an inequality. It helps illustrate the solution set and quickly convey information about which numbers satisfy the inequality.
To graph our example, \((-2, 1]\), follow these steps:

• Draw a horizontal line to represent the number line.
• Mark significant points, like \(-2\) and \(1\), on the line.
• Use circles to denote whether these points are included in the solution set or not.
• Shade the region on the line between the appropriate points to indicate all numbers that are part of the solution set.

This kind of visual diagram uses open and closed circles as symbols, which we will discuss in more detail in later sections. A number line graph provides an immediate visual indication of all numbers that satisfy a given inequality.

Solution Set

The solution set of an inequality is the range or collection of values that satisfy it. For our specific inequality, \(-2 < x \leq 1\), the solution set contains all real numbers \(x\) such that \(-2 < x \leq 1\).
The solution set is represented in interval notation, a mathematical shorthand that conveys the same information quickly. Our solution set is denoted as \((-2, 1]\), using parentheses and brackets to indicate whether endpoints are included.

• Parentheses \(()\) mean the endpoint is not included (open).
• Brackets \([]\) mean the endpoint is included (closed).

Thus, our solution set begins at any number just greater than \(-2\) and ends at \(1\), including \(1\). This notation efficiently communicates the range of possible solutions, which can then be further analyzed or graphed.

Open and Closed Circles

When graphing inequalities on a number line, open and closed circles are used as markers to indicate whether endpoints are part of the solution set.

• Open circles are used to show that the number at that point is not included in the solution set. For instance, in the inequality \(-2 < x\), the point \(-2\) is marked with an open circle because \(x\) is greater than \(-2\), but not equal to it.
• Closed circles indicate that the endpoint is included in the solution set. In \(x \leq 1\), the point \(1\) is denoted with a closed circle, indicating that \(x\) can equal \(1\).

This graphical representation is crucial because it instantly shows which values are part of the solution set and which are not. By understanding the use of open and closed circles, you can accurately interpret and create number line graphs for various inequalities.