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Solving inequalities helps us find the range of values a variable can take to satisfy a given condition. In the case of absolute value inequalities, we solve them by considering what absolute value represents. The inequality \(|x-9| \geq 2\) tells us that the distance between the number \(x\) and 9 is 2 units or more. To solve it, we break it into two scenarios: the distance can be to the right of 9 or to the left.

• When solving \(x - 9 \geq 2\), we add 9 to both sides to find \(x \geq 11\).
• For \(x - 9 \leq -2\), we again add 9 to both sides, resulting in \(x \leq 7\).

The solution to the original inequality thus comprises these two conditions combined with an "or." This means any value of \(x\) that satisfies either condition is part of the solution set. This process simplifies the seemingly complex inequality into straightforward linear inequalities.

Interval notation is a mathematical shorthand used to describe the solution set of inequalities in a concise manner. It's particularly useful when dealing with absolute value inequalities as it presents intervals compactly. In our example, the solution to the inequality \(|x-9| \geq 2\) is a union of two intervals: \((-fty, 7]\) and \([11, fty)\).

• \((-fty, 7]\) includes all numbers less than or equal to 7.
• \([11, fty)\) includes all numbers greater than or equal to 11.

The union symbol \(\cup\) denotes that we are considering all values in either interval. This method saves space and provides a clear view of where the solutions lie on the number line. Interval notation helps us quickly communicate the solutions without listing all possible numbers.

Graphing solutions on a number line visually represents the solutions to inequalities, making it easier to understand their range and inclusivity. For \(|x-9| \geq 2\), we graph by marking and shading the interval portions where our solutions lie.

• Start by marking a solid dot at 7 to include it and shade to the left towards \(-\infty\).
• Then, mark a solid dot at 11 and shade to the right stretching towards \(\infty\).

The solid dots indicate that the endpoints are part of the solution, thanks to the 'or equal to' part of the inequality. This straightforward graph shows that all numbers less than or equal to 7 and all numbers greater than or equal to 11 satisfy the inequality. Visuals like this help students better grasp mathematical concepts, as they clearly see which values meet the inequality condition.

The concept of absolute value gives us a way to measure the distance of a number from zero on the number line, regardless of direction. In inequalities, this becomes very useful. For instance, the inequality \(|x-9| \geq 2\) essentially asks: "How far is \(x\) from 9, irrespective of the direction?" This distance must be at least 2 units.

• Absolute value ensures this measurement is always positive or zero—never negative.
• The inequality takes into account both possible directions from the specified point.

Understanding absolute value is key to correctly setting up these kinds of inequalities. It helps break down a seemingly complex expression into manageable parts. Knowing this allows students to translate and solve absolute value inequalities into simpler linear forms easily.