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Chapter 1: Problem 11

The compound inequality \(a

Short Answer

Expert verified
a < x < b

Step by step solution

01

Identify the Compound Inequality Components

Recognize the given compound inequality is composed of two separate inequalities: \(a < x\) and \(x < b\).
02

Combine the Inequalities

Combine the two inequalities into a single three-part inequality by placing the same variable \(x\) between the two parts. This gives us: \[a < x < b\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Notation
Inequality notation is a way to express that one quantity is less than or greater than another. We use symbols like <, >, 鈮�, and 鈮�. These symbols help show the relationship between numbers or variables. For example, the notation \(a < x\) means that \(x\) is greater than \(a\). Similarly, \(x 鈮� b\) means \(x\) is less than or equal to \(b\). By using these notations, we can describe a range of possible values for variables.
Three-Part Inequality
A three-part inequality is a special type of compound inequality, where you link two inequalities to describe a range for a variable. Instead of writing two separate inequalities, we combine them into one statement. For example, if we have \(a < x\) and \(x < b\), we can write it as \ [a < x < b] \ Here, \(x\) is greater than \(a\) and less than \(b\). This shows all the possible values \(x\) can take that satisfy both conditions. It's a simpler way to express ranges.
Inequality Solving Steps
To solve compound inequalities, follow these steps:

  • **Identify Components**: First, look at the individual inequalities given. For instance, \(a < x\) and \(x < b\).

  • **Combine Inequalities**: Place the common variable between both parts of the inequality. This gives us \ [a < x < b] \

  • **Simplify and Check**: Ensure there are no contradictions and that the combined inequality makes sense. This helps confirm the range of possible values for the variable.

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Most popular questions from this chapter

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