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Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols; it is a language through which we describe patterns and relationships between numbers and variables. In the context of our problem, algebra guides us through the process of solving inequalities. Like a balancing act, algebra insists on maintaining equality, or in the case of inequalities, the direction of inequality, by performing the same operation on both sides.

In the given exercise, algebraic operations such as addition and division are used to simplify the inequality and isolate the variable. When we add or subtract the same number from both sides of an inequality, the direction of the inequality remains the same. However, when we multiply or divide both sides by a negative number, the inequality sign must be flipped to preserve the inequality's truth. This is a fundamental rule in algebra that is paramount to solving inequalities correctly.

Inequality notation is used to represent the relationship between expressions that are not equal. Unlike equations, where two expressions are equal, inequalities show that one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used include < for 'less than', > for 'greater than', \( \leq \) for 'less than or equal to', and \( \geq \) for 'greater than or equal to'.

When presenting the final answer, it's crucial to use proper notation to convey the solution set. In the solved exercise, the solution is written in inequality notation as \( x < 7 \), which reads 'x is less than 7'. Additionally, interval notation \( (-\infty, 7) \) is introduced as a concise way to represent all the numbers that are less than 7. It is essential to familiarize oneself with these notations as they are widely used to express solutions to inequalities.

To isolate the variable means to rearrange an algebraic expression so that the variable we're solving for is on one side of the equation or inequality by itself. The goal is to get the variable alone on one side and everything else on the other to determine its value or range of values.

In our exercise, we start with \(6x - 11 < 31\) and aim to solve for \(x\). The first step involves adding 11 to both sides to remove the constant term from the side containing the variable. This leaves us with \(6x < 42\). Next, we divide both sides by 6 to get the variable \(x\) by itself, yielding the solution \(x < 7\).

Why Isolating the Variable Matters

Isolating the variable is critical as it clearly defines the boundaries of the solution set. In inequalities, it shows us the entire set of numbers that satisfy the given condition, providing a clear picture of all possible solutions.