Problem 90 - Intermediate Algebra

Question

Explain why the double inequality \(2

Step-by-Step Solution

Answer

The double inequality \(2 < x < 8\) means \(x\) is greater than 2 and less than 8, which can be expressed as \(2 < x\) and \(x < 8\) separately.

Step 1: Understanding the Concept of Double Inequality

A double inequality like \(2 < x < 8\) is a compact way of expressing two separate inequalities that must both be satisfied simultaneously. It means \(x\) is greater than 2 and, at the same time, less than 8.

Step 2: Breaking Down the Double Inequality

The double inequality \(2 < x < 8\) can be separated into two distinct inequalities: \(2 < x\) and \(x < 8\). These two conditions together restrict \(x\) to be between 2 and 8, excluding the endpoints.

Step 3: Expressing the Separate Conditions

To write the double inequality as two separate inequalities, we write: \(2 < x\) which implies \(x\) is greater than 2, and \(x < 8\) which implies \(x\) is less than 8. The logical 'and' between them specifies that both conditions must be true for any value of \(x\).

Step 4: Equivalence of the Two Forms

Both the double inequality \(2 < x < 8\) and the compound statement \(2 < x\) and \(x < 8\) convey the same set of permissible values for \(x\). They both define the range of \(x\) such that the value is strictly between 2 and 8.

Key Concepts

Compound Inequalities

Compound inequalities are a way to express two or more inequalities that are combined into a single statement.
They are used to show that a variable, such as \(x\), satisfies several conditions at once. This means it lies within a specific range or set of values.

• A double inequality like \(2 < x < 8\) is one example of a compound inequality.
• It conveys two separate conditions: \(x > 2\) and \(x < 8\).
• Each condition in the inequality must be true for any potential value of \(x\).

Breaking a double inequality into its separate parts helps in clarity and understanding. It's essentially spelling out that \(x\) must meet both criteria, one right after the other. This approach ensures we are aware of the full scope of where \(x\) can be positioned on the number line.

Inequality Notation

Inequality notation is a symbolic way to represent expressions that describe the relative size or order of two values.
Different symbols are used to show how these values compare.

• \(<\) and \(>\) denote "less than" and "greater than," respectively.
• A double inequality uses these symbols to form a bridge between two potential conditions on a variable.

In the expression \(2 < x < 8\), inequality notation allows us to elegantly express two conditions in one compact form. Each part works independently yet together, delivering clear constraints.
By using notation efficiently, such expressions demonstrate exactly where a number should fall in a range.

Logical Conjunction

Logical conjunction involves combining multiple logical statements, each typically having a Boolean value (true or false). It requires all components to be satisfied for the overall expression to be true.

• The logical 'and,' often represented by \(\land\), signifies this combination.
• In the case of double inequalities like \(2 < x < 8\), the statement is equivalent to \(2 < x\) and \(x < 8\).
• Both conditions must be true for \(x\) to be within the specified range.

By writing inequalities this way, we clearly state the necessity of satisfying all parts.
For example, with \(2 < x\) and \(x < 8\), \(x\) must not only be greater than 2 but also less than 8. If either part fails, the entire statement becomes false.
This logical approach assures strict adherence to requirements and helps prevent mathematical errors.