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When solving inequalities, especially those involving absolute values, it’s important to understand that absolute values represent the distance of a number from zero on the number line. This means that the expression inside the absolute value can be both positive and negative, and we must consider both possibilities to find all solutions for the inequality.

The original inequality given was \( \left|\frac{3x-5}{6}\right|>5 \). This can be split into two separate inequalities:

• \( \frac{3x-5}{6} > 5 \)
• \( \frac{3x-5}{6} < -5 \)

Each inequality is solved independently, and the solutions are combined to capture the full range of values that \( x \) can take. Solving these inequalities involves basic algebraic manipulations like multiplication, addition, and division to isolate \( x \) on one side.

Interval notation is a concise way of expressing a set of solutions for inequalities. It shows the range of numbers that satisfy the inequality by using brackets and parentheses to indicate where numbers are included or excluded.

In the solution to the original inequality, we found \( x \) values where \( x > \frac{35}{3} \) or \( x < -\frac{25}{3} \). In interval notation, these two conditions come together to express the complete solution. The intervals are written as:

• \( (-\infty, -\frac{25}{3}) \)
• \( (\frac{35}{3}, \infty) \)

The \( \cup \) symbol, which means "union," is used to join these two intervals, indicating that any \( x \) value within either of these ranges will satisfy the original inequality. Parentheses are used to denote that the endpoints are not included, adding a clear understanding of open intervals.

Algebraic manipulation is a key skill needed to solve inequalities, especially those involving fractions and absolute values. The goal is to simplify the expression and isolate the variable \( x \).

For instance, to tackle the inequality \( \frac{3x-5}{6} > 5 \), we first multiply both sides by 6 to eliminate the fraction, resulting in \( 3x - 5 > 30 \). Next, we simplify further by adding 5 to both sides, yielding \( 3x > 35 \). Finally, to solve for \( x \), divide both sides by 3, leading to \( x > \frac{35}{3} \).

These steps need careful attention to maintaining the inequality throughout. This means being very precise with operations such as multiplication and division, ensuring the inequality sign is flipped correctly especially when multiplying or dividing by negative numbers. However, in this case, since we only dealt with positives, the inequality sign remained the same. The strategies used are essential for accurately solving and understanding a wide variety of algebraic problems.