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Interval notation is a concise way of representing a range of numbers. It's particularly useful when working with solutions to inequalities. The key components are the parentheses or brackets:

• Parentheses, like \((-10, -9)\), indicate that the endpoints are not included in the set. This is also known as an "open interval."
• Brackets, on the other hand, like \([-10, 9]\), would indicate that the endpoints are included, which is referred to as a "closed interval."

In our exercise, for the inequality \-10 < x < -9\, since the values at the endpoints are not part of the solution, we use parentheses for both -10 and -9. Thus, it is written as \((-10, -9)\).
Understanding and using interval notation correctly helps in communicating solutions clearly and efficiently.

Graphical representation involves plotting the solution of inequalities on a number line. This visual representation helps in understanding the solution better by showing all possible values that satisfy the inequality. To graph the solution of \((-10, -9)\), you follow these steps:

• Draw a horizontal line representing the number line.
• Mark the points \(-10\) and \(-9\) with open circles. Open circles mean these points are not included in the solution set.
• Shade the segment between \(-10\) and \(-9\) to illustrate that this is the range of values satisfying the compound inequality.

This method not only aids in understanding but is essential for portraying inequalities succinctly and graphically.

Solving inequalities involves finding all values of a variable that make an inequality true. Similar to solving equations, but with some key differences, especially when multiplying or dividing by negative numbers. Let's look at the process:1. **Isolate the variable**: Like solving equations, the goal is to get the variable on one side.

• Example: For \(2.2x < -19.8\), divide both sides by 2.2 to find the values of \(x\).

2. **Reverse the inequality**: When dividing or multiplying an inequality by a negative number, the inequality sign must be flipped.

• Example: For \(-4x < 40\), divide by -4 to isolate \(x\), resulting in the inequality sign changing direction to \(x > -10\).

3. **Combine solutions for compound inequalities**: If the compound inequality uses "and," find the intersection of the solution sets.

• Example: Combine \(x < -9\) and \(x > -10\) to find where they overlap: \(-10 < x < -9\).

Solving compound inequalities is a fundamental skill that allows one to deal with real-world scenarios represented by inequalities.