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Understanding interval notation is crucial when working with inequalities, as it provides a clear and concise way to describe sets of numbers that satisfy the inequality's condition. It is a shorthand for communicating which numbers fall within a certain range, often used to express the solution set to an inequality. In interval notation, we use parentheses () to denote that an endpoint is not included, also known as an open interval, and square brackets [] to denote that an endpoint is included, known as a closed interval. For example, the notation \( (a, b) \) denotes all the numbers greater than \( a \) and less than \( b \). If the range extends indefinitely in one direction, we use infinity symbols: \( -\infty \) for numbers decreasing without bound, or \( \infty \) for numbers increasing without bound. Therefore, in our exercise, the notation \( (-\infty, 4) \) indicates all real numbers that are less than 4, since 4 is not included in the solution set.

Isolating the variable is a fundamental step in solving both equations and inequalities. The goal is to get the variable by itself on one side of the inequality sign so we can see what values it can take. To isolate the variable, we perform basic algebraic operations like addition, subtraction, multiplication, or division, always remembering to do the same operation on both sides of the inequality to maintain balance. When dealing with inequalities, there’s an additional rule one must remember: if you multiply or divide both sides by a negative number, you must flip the inequality sign. This step is about simplifying the given inequality to its most basic form, allowing us to easily identify the set of values for the variable. In our example, we added 4 to both sides to isolate \( x \) and were left with the simplified inequality \( x < 4 \) from the original \( x - 4 < 0 \).

The solution to an inequality is the set of all possible values of the variable that make the inequality true. Unlike equations, where the solution is a single number, inequalities typically have a range of solutions. After isolating the variable, the next step is to interpret the inequality to determine the solution set. With the inequality \( x < 4 \) from our exercise, we understand that any number less than 4 will satisfy the inequality. The solutions are not just whole numbers; they include all real numbers that are less than 4. When translating this into interval notation, we use \( (-\infty, 4) \) to express this continuous range of numbers, reinforcing the concept that there is no upper limit to how small the numbers can be, and that 4 itself is not a solution. This interval represents all the solutions to the inequality, and it is an essential way to convey this information in a precise mathematical format that is universally understood.