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When expressing the solution set of an inequality, interval notation is a useful method. It allows us to describe the set of numbers that satisfy the inequality in a concise manner. For instance, in the inequality \(t \leq -\frac{10}{9}\), the solution set includes all numbers that are less than or equal to \(-\frac{10}{9}\).
To write this in interval notation:

• Use parentheses \(()\) to denote that an endpoint is not included (open interval).
• Use brackets \([]\) to indicate that an endpoint is included (closed interval).

The solution \(t \leq -\frac{10}{9}\) includes \(-\frac{10}{9}\) itself and all numbers smaller, hence we use a bracket on that side and write it as \((-\infty, -\frac{10}{9}]\). The negative infinity symbol \((-\infty)\) always uses a parenthesis because infinity is not a specific number and cannot be reached.
Using this approach efficiently communicates the entire solution set at a glance.

The solution set of an inequality is the range of all possible numbers that satisfy the inequality condition. When dealing with inequalities in equations, it's important to clearly understand what numbers belong to the solution set.
For instance, the original inequality \(-9t + 6 \geq 16\) needs to be solved to determine which values of \(t\) make the inequality true. Step by step, you isolate the variable until you reach a simplified inequality \(t \leq -\frac{10}{9}\). This tells us that every number less than or equal to \(-\frac{10}{9}\) is part of our solution set.
Understanding the solution set ensures that you include every possible value that makes the inequality true and exclude those that do not.

Graphing inequalities on a number line provides a visual representation of the solution set. It helps in understanding which values are included in the solution and which are not.
When graphing the inequality \(t \leq -\frac{10}{9}\):

• Draw a number line and locate the point \(-\frac{10}{9}\).
• Since the inequality is "less than or equal to," place a closed circle on \(-\frac{10}{9}\). The closed circle indicates that this number is part of the solution set.
• Shade the region to the left of \(-\frac{10}{9}\) to show that all smaller numbers are included in the solution.

This visual method makes it easier to interpret inequalities, especially when dealing with more complex sets of numbers. It clearly displays the inclusive or exclusive nature of boundary points when solving inequalities.