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Interval notation is a concise way to describe a range of numbers that are solutions to an inequality. It uses parentheses and brackets to indicate whether endpoints are included or excluded from the set. In the inequality \( t > 11 \), we are not including 11, which means we use a parenthesis. Therefore, the interval starts just above 11 and goes to infinity, represented as \((11, \infty)\).
Here are some key points:

• Use parentheses \(()\) if the endpoint is not included in the set (like 11 in this example).
• Use brackets \([]\) if the endpoint is included.
• Infinity \(\infty\) or negative infinity \(-\infty\) are always accompanied by a parenthesis because you can't reach actual infinity.

Interval notation helps provide a clear, mathematical description of all values that satisfy our inequality, making it easy to communicate and interpret solutions.

Graphing inequalities involves representing the solution set of an inequality on a number line. This visual representation provides a clear way to see which numbers satisfy the inequality and how they relate to each other.

Steps for Graphing

• Identify the boundary number, which here is 11.
• Determine whether the boundary is included or not. For \( t > 11 \), use an open circle at 11, indicating 11 itself is not part of the solution.
• Shade or draw a line to the right of the circle, indicating that all numbers greater than 11 satisfy the inequality.

This simple graphing technique shows all solutions in one quick, visual sweep. Open and closed circles effectively indicate whether boundary values are included or not, making graph comprehension very intuitive.

Isolation of variables is a fundamental skill in solving inequalities and equations alike. The goal is to get the variable by itself on one side of the inequality to determine its possible values.

How to Isolate Variables

• Perform the same operation on both sides of the inequality, like addition, subtraction, multiplication, or division. In \( 2t > 22 \), dividing both sides by 2 isolates \( t \).
• Always ensure you perform inverse operations to "undo" existing operations around the variable.
• If you multiply or divide by a negative number, remember to reverse the inequality sign.

Isolating variables correctly ensures the solution accurately represents all possible values that satisfy the inequality. It’s a vital step in forming the solution set.