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When solving inequalities, one common approach is the addition principle. This principle states that the same number can be added or subtracted from both sides of an inequality without changing the inequality's direction. This method helps maintain the balance and truth of the statement. For example, in the inequality \( y + \frac{1}{3} \leq \frac{5}{6} \), to isolate the variable \( y \, \text{on the left-hand side} \), we use the addition principle and subtract \( \frac{1}{3} \) from both sides. This keeps the inequality valid and balanced, making it simpler to solve.

Working with fractions in inequalities often requires finding a common denominator to simplify the expressions. A common denominator is a shared multiple of the denominators in the fractions involved, which allows you to add or subtract them more easily. For instance, to solve \( y + \frac{1}{3} \leq \frac{5}{6} \), we need to subtract \( \frac{1}{3} \). To do this, we first convert \( \frac{1}{3} \) into a fraction with a denominator of 6, because the original inequality's right-hand side already has 6 as the denominator: \( \frac{1}{3} = \frac{2}{6} \). This conversion ensures that the fractions can be easily subtracted: \( \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \). Using a common denominator simplifies calculations and makes it easier to understand and solve inequalities.

Isolating the variable is a crucial step in solving any equation or inequality. This process involves manipulating the inequality so that the variable stands alone on one side. For the inequality \( y + \frac{1}{3} \leq \frac{5}{6} \), we isolate \( y \), \text{by subtracting the fraction} \( \, \frac{1}{3} \) \text{from both sides using the addition principle}. This gives us: \( y + \frac{1}{3} - \frac{1}{3} \leq \frac{5}{6} - \frac{1}{3} \), and simplifies to \( y \leq \frac{1}{2} \). By isolating the variable \( y \), we make it clear what values \( y \) can take, which is all values less than or equal to \( \frac{1}{2} \). Ensuring that the variable is isolated helps in clearly understanding the range of possible solutions.