In Exercises 51–58, solve each compound inequality. 6 < x + 3 < 8

Key Concepts

Compound Inequalities

Compound inequalities involve two inequalities that are combined into one statement. They can be expressed in the form 'a < x < b', indicating that x must satisfy both conditions simultaneously. Understanding how to manipulate and solve these inequalities is crucial for finding the range of values that satisfy the entire statement.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve inequalities. This involves performing operations such as addition, subtraction, multiplication, or division on both sides of the inequality to get the variable alone. In the context of compound inequalities, this means applying the same operations to all parts of the inequality to find the solution set.

Solution Set

The solution set of an inequality is the set of all values that satisfy the inequality. For compound inequalities, the solution set is often expressed in interval notation, which provides a concise way to represent the range of values. Understanding how to interpret and express solution sets is essential for effectively communicating the results of solving inequalities.