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When expressing the solution of a linear inequality, we often use interval notation. This is a concise way of writing down the set of numbers that form the solution. In interval notation, we describe parts of the number line that include all possible solutions.
- **Use Parentheses,** like \((\) or \()\), when the endpoints are not included in the set. This corresponds to strict inequalities such as \( < \) or \( > \).
- **Use Brackets,** like \([\) or \(]\), when the endpoints are part of the solution set. This is typical for \( \leq \) or \( \geq \) relations.
For the inequality \( y < 1 \), we express this in interval notation as \( (-\infty, 1) \). The parentheses indicate that 1 is not included. It means all numbers less than 1 are solutions. Because \(-\infty\) is a concept, not a number, we always use a parenthesis with it.

Simplifying expressions is a crucial step in solving inequalities, especially when they involve fractions or more complex terms. The process of simplification helps to make the problem more manageable and clear.
- **Distribution:** This involves multiplying through any parentheses to simplify an expression. For example, \(-\frac{1}{2}(5-y)\) becomes \(-\frac{5}{2} + \frac{1}{2}y\).
- **Working with Fractions:** It's important to convert fractions so that they have a common denominator when combining or comparing them. Often, this involves identifying the least common denominators and adjusting terms as needed to simplify the expression fully.
The purpose of simplifying expressions is to streamline the problem to easily spot the steps needed to isolate the variable and solve the inequality.

Combining like terms is a method used to simplify expressions further by gathering similar terms together. This helps reduce clutter and can provide a clearer path toward solving the inequality.
- **Identify Similar Terms:** Terms are "like" if they have the same variable raised to the same power. In the exercise, terms like \( \frac{2}{3}y \) and \( \frac{1}{2}y \) can be combined because they both include the variable \( y \).
- **Addition and Subtraction:** This involves adding or subtracting coefficients of like terms. For example, \( \frac{2}{3}y + \frac{1}{2}y \) becomes \( \frac{4}{6}y + \frac{3}{6}y \), which simplifies to \( \frac{7}{6}y \).
Combining like terms streamlines the equation, making it easier to see how to isolate the variable for the next steps in solving the inequality.