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When expressing solutions to inequalities, interval notation offers a concise way to represent the range of possible values. In our problem, after solving the compound inequality, we found that the variable \(x\) is greater than 11 and less than 17. In interval notation, this is written as \((11, 17)\). This reads as "\(x\) is between 11 and 17, not including 11 and 17 themselves."

Interval notations use parentheses \(( )\) to show that endpoints are not included (also known as open intervals), and brackets \([ ]\) to indicate that endpoints are included (closed intervals). Here are some examples to illustrate:

• \((a, b)\) means \(a < x < b\)
• \([a, b]\) means \(a \leq x \leq b\)
• \((a, b]\) means \(a < x \leq b\)
• \([a, b)\) means \(a \leq x < b\)

By using these notations, we can easily convey the range of solutions in a standardized form that is widely understood in mathematics.

Solving compound inequalities involves finding the range of values that satisfy all parts of the inequality. In our example, we dealt with the inequality \(5 < x - 6 < 11\). This means we're looking for the values of \(x\) where \(x - 6\) is simultaneously greater than 5 and less than 11.

The process of solving such inequalities can be broken down into steps:

• Identify the inequality type: A compound inequality involves two inequality signs and links conditions together.
• Isolate the variable: Perform operations like addition or subtraction on all parts of the inequality to solve for the variable. Always ensure that the operations maintain the inequality balance.
• Write the solution: After isolating the variable, express the solution set using inequality notation first, then in interval notation.

By ensuring each step is done methodically, you solve the inequality effectively.

Understanding compound inequalities is an essential part of mathematics education. These concepts not only teach logical thinking but also improve problem-solving skills. Teachers and educators often use various tools to help students grasp these ideas, such as visual aids and analogies.

To help students understand:

• Use number lines to visually demonstrate how inequalities span specific ranges.
• Encourage students to solve each part of a compound inequality separately before combining results for the full solution.
• Provide plenty of practice problems to build familiarity and confidence.

Mathematics education is about building a solid foundation of concepts that can be applied to various real-world situations. By engaging with inequalities, students develop critical skills that are useful beyond the classroom.