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Interval notation is a way to describe sets of numbers along a number line. It provides a neat and organized method to express the range of solutions for inequalities.

When we solved the inequality \(\frac{1}{2}(x-4) > x+8\) and found that \(x < -20\), we needed to write this solution in interval notation.

Here’s how it works:

• Use a parenthesis \(()\) or bracket \([]\) to describe the endpoints of intervals.
• Parentheses \(()\) mean that the endpoint is not included.
• Brackets \([]\) mean that the endpoint is included.

The solution set for \(x < -20\) in interval notation is written as \((-∞, -20)\). This means that while \-20\ itself isn't part of the solution, everything less than \-20\ is.

Notice the negative infinity symbol \(-∞\); it indicates that the interval extends indefinitely in the negative direction. Always remember to pair \(-∞\) or \(∞\) with a parenthesis since infinities are never actual endpoints.

Graphing inequalities helps visualize the solution. Let’s graph the solution set for \(x < -20\).

Here’s how it's done:

• Draw a number line.
• Locate and mark the number \-20\ on the line.
• Because \-20\ is not included in the solution, draw an open circle at \-20\ to indicate it’s not part of the solution.
• Shade the region to the left of \-20\ to represent all numbers less than \-20\.

By following these steps, the graph clearly shows the range of numbers that satisfy the inequality. The open circle and shaded region visually affirm that our solution includes all numbers less than \-20\ but not \-20\ itself.

Isolating the variable is a fundamental process in solving inequalities. This involves rearranging the inequality to get the variable by itself on one side of the inequality sign. Let’s look at our inequality: \(\frac{1}{2}(x-4) > x + 8\).

Here are the steps:

• Expand the inequality: We distributed the \( \frac{1}{2} \) to each term inside the parentheses, simplifying to \( \frac{1}{2}x - 2 > x + 8 \).
• Combine like terms: We subtracted \( x \) from each side, combining the x-terms on one side: \( \frac{1}{2}x - x - 2 > 8 \ => -\frac{1}{2}x - 2 > 8 \).
  
  
• Isolate the variable: Adding 2 to both sides resulted in \( -\frac{1}{2}x > 10 \). Finally, we multiplied by \(-2\) and remembered to reverse the inequality sign, arriving at \( x < -20 \).

Isolating the variable step-by-step simplifies complex inequalities and prepares them for graphing or interval notation. Always ensure to reverse the inequality sign when multiplying or dividing by a negative number during isolation.