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Interval notation is a way of representing a set of numbers between two endpoints on the number line. It is a concise method to express inequalities and is particularly useful when the solutions involve ranges. The endpoints are included or excluded using brackets:

• Round brackets, like ")" and "(", indicate that the endpoint is not included in the set (also known as an "open interval").
• Square brackets, like "]" and "[", signify that the endpoint is included (known as a "closed interval").

In the exercise, for inequality part (a) where we found that the solution was all numbers greater than 8, we expressed this in interval notation as (8, \infty). Here, the round bracket at 8 means 8 is not included, and \(\infty\) signifies that there is no upper limit.
For part (b), where the solution was all numbers less than or equal to 8, the interval notation is \((-\infty, 8]\). The square bracket at 8 includes 8 in the solution set.

A solution set is the collection of all possible solutions that satisfy a given inequality or equation. It represents all the valid values of a variable that make the inequality true. For example, in this exercise,

• For inequality \(9 - (x + 1) < 0\), the solution set is the set of all real numbers greater than 8, which we expressed in the solution format as \(x > 8\).
• For inequality \(9 - (x + 1) \geq 0\), the solution set includes all real numbers less than or equal to 8, expressed as \(x \leq 8\).

Understanding solution sets allows us to know all possible scenarios meeting the criteria of the inequality. They are crucial in various applications, such as optimization problems and real-life scenarios, where constraints dictate possible outcomes.

A graphical solution to an inequality involves drawing the solution set on a number line. This visual representation helps in easily understanding the range of solutions and whether or not certain values are included.
For the solutions in our exercise:

• For the inequality \(x > 8\), we would draw an open circle at 8 and shade the line extending to the right towards infinity. This illustrates that the numbers are greater than 8 but do not include 8 itself.
• For \(x \leq 8\), a filled-in circle at 8 indicates that 8 is part of the solution set, with shading to the left towards negative infinity, showing the values are 8 and below.

Using graphical solutions offers a quick check to ensure logical consistency and correctness while providing a visual aid that complements the analytical method.

Analytical methods involve using algebraic manipulation to solve inequalities or equations precisely. In solving inequalities, we follow a systematic approach similar to solving equations but with special care regarding the inequality sign.
Here is a brief overview of this approach as used in the given exercise:

• We began by simplifying the inequalities \(9 - (x + 1) < 0\) and \(9 - (x + 1) \geq 0\) to \(-x + 8 < 0\) and \(-x + 8 \geq 0\), respectively, by distributing the negative sign and simplifying the expression.
• Next, we isolated the variable \(x\) by performing arithmetic operations. We subtracted 8 from both sides, followed by multiplying through by -1. It is important to remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
• Finally, this led to solutions \(x > 8\) and \(x \leq 8\), giving us the full range of solutions for both inequalities.

Analytical methods ensure a step-by-step logical process, leading to accurate and reliable solutions supported by mathematical reasoning.