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When dealing with inequalities, expressing the set of solutions can be efficiently achieved using interval notation. This system is a shorthand way of describing intervals on the number line, capturing all the numbers that fall between two endpoints. It is crucial when conveying the range of solutions for inequalities.

To master interval notation, we first acknowledge two forms of intervals: open and closed. Open intervals, denoted by parentheses \( ( ) \), signify that the endpoints are not included in the set. Conversely, closed intervals, denoted by brackets \[ [ ] \], include the endpoints. For the given exercise, the solution in interval notation is \(x \in (-3,3)\), which means all real numbers greater than -3 and less than 3, but not including -3 and 3 themselves.

Writing Intervals Correctly

To ensure a correct usage of interval notation in your solution, always start with the lower number, followed by a comma, and then the higher number. Additionally, use infinity \( \infty \) or negative infinity \( -\infty \) with open parentheses to indicate that an interval extends indefinitely in either the positive or negative direction.

In solving inequalities, one of the fundamental steps is isolating the variable. This is the process of manipulating the inequality to have the variable on one side and all other terms on the opposite side.

Let's break it down using the given exercise. Initially, the variable \(x\) is within a fraction. By multiplying all terms of the inequality by 2, the first step has neatly removed the denominator, simplifying our inequality to \(0 < x + 3 < 6\). Next, to isolate \(x\), we must ensure it stands alone. By subtracting 3 from all parts of the inequality (Step 2), \(x\) is successfully isolated and we have \( -3 < x < 3\).

Why Isolating the Variable Matters

Isolating the variable is crucial as it makes the solution of the inequality explicit, allowing for a clear understanding of the set of all possible values for \(x\) that satisfy the inequality. This step paves the way for the final solution, which can be represented using interval notation.

Effective inequality manipulation is a key skill when solving inequalities. It involves applying arithmetic operations and logic to both sides of the inequality to maintain its true balance, all while aiming to isolate the variable.

In our example, initially, both multiplication by 2 and subtraction of 3 were used to manipulate the inequality. This action adheres to the fundamental principle that what you do to one part of an inequality, you must do to all parts to maintain its truth value.

Remember the Rules

Multiplying or dividing by a positive number preserves the direction of the inequality. However, multiplying or dividing by a negative number reverses the inequality's direction. It's also important to never assume that the inequality will always be 'less than' or 'greater than' throughout the process; always reassess the inequality's direction after each manipulation step to avoid errors.