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Understanding the concept of distance on the real line is crucial for tackling problems involving absolute value inequalities. When we talk about the distance of a point from zero on the real line, we're referring to how far that point is from the origin, without considering direction. This distance is measured as an absolute value. So, an expression like \(|x|\) reflects the distance of point \( x \) from zero.

In part (a) of the exercise, when we say a distance "less than 3," we are referring to points located within 3 units of zero. On the real line, this means any number between -3 and 3. In mathematical terms, for a distance less than 3, we write \(|x| < 3\).

Conversely, for part (b), when we describe a distance "greater than 3," it represents points that lie beyond 3 units away from the origin, either to the left or the right of zero. For this scenario, the absolute value inequality is written as \(|x| > 3\). This captures two possible regions: \( x > 3 \) or \( x < -3 \).

In simpler words, understanding the concept of real line distance involves picturing a number line and recognizing which sections of that line meet the specified distance criteria from zero.

Solving inequalities involves finding a set of values that satisfy given conditions. Absolute value inequalities might initially seem complex, but they can often be broken down into two basic inequalities. This process starts by grasping what the inequality means on a conceptual level.

For example, let's examine the inequality \(|x| < 3\) from part (a). This states that the distance from zero should be less than 3. We translate this into two inequalities: \(-3 < x < 3\). Here, \( x \) must lie between -3 and 3, excluding -3 and 3 themselves.

In part (b), the inequality \(|x| > 3\) suggests the distance should be greater than 3, translating to \( x > 3 \) or \( x < -3 \). These separate inequality solutions indicate values of \( x \) that are farther than 3 units from zero, in either direction.

To solve absolute value inequalities effectively, follow these steps:
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• Understand the absolute value expression and the conditions given.
• Break it into "less than" or "more than" scenarios.
• Convert these scenarios into individual math inequalities.
• Determine the values that meet each inequality.

\)Through these steps, we can solve complex inequality problems with ease and clarity.

Understanding the properties of absolute values is essential for correctly interpreting and solving absolute value inequalities. These properties tell us that \(|x|\) will always give a non-negative outcome, regardless of whether the input \(x\) is a negative or positive number. This is why dealing with absolute values often involves two cases.

The primary properties of absolute values include:
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• \(|x| = x\) if \(x\geq 0\)
• \(|x| = -x\) if \(x < 0\)

\)
These properties ensure that the result is indeed a distance, which can only be zero or positive.

In the context of the exercise, these properties help explain the split in reasoning when we solve inequalities like \(|x| < a\) and \(|x| > a\). For \(|x| < a\), we set up the inequality as \(-a < x < a\), capturing numbers that fall within a range centered at zero. And for \(|x| > a\), the inequalities \(x > a\) or \(x < -a\) describe numbers that extend beyond that specified distance from zero.

By leveraging these properties, absolute value inequalities can be solved by understanding the geometric interpretation of distance and applying logical reasoning to break them into simpler parts.