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When solving absolute value inequalities, such as \(|3 + x| > 7\), it's important to understand how to express solution sets using interval notation. This notation is a mathematical language to describe a range of values easily. It uses brackets and parentheses to indicate which endpoints are included in the set.

- **Parentheses** \(( , )\) are used to denote that an endpoint is **not** included. You might hear this called an 'open interval'.
- **Brackets** \([ , ]\) are used when the endpoint **is** included, known as a 'closed interval'.

For our example, \(x > 4\) and \(x < -10\) translate to interval notation as \((4, \,\infty)\) and \((-\infty, \, -10)\), respectively. The symbol \(\cup\) is used to denote the union of two intervals. This means we're combining both sets of solutions. Our complete solution in interval notation becomes \((-\infty, \, -10) \cup (4, \, \infty)\). You'll notice the parentheses signify that -10 and 4 are not solutions, matching our inequality's signs.

A number line graph is a visual representation that depicts the solution set of an inequality. It's a useful tool for visual learners to quickly see which values satisfy the inequality.

To graph a solution set like \((-\infty, -10) \cup (4, \infty)\), follow these steps:

• Draw a horizontal line to represent real numbers.
• Mark numbers -10 and 4 on your line. These numbers are **not** included in the solution set, so use open circles to represent them.
• For \(-10\), shade the line to the left, indicating that all numbers less than -10 satisfy the inequality \(x < -10\).
• For 4, shade the line to the right, showing that all numbers greater than 4 satisfy \(x > 4\).

This graph represents that any number less than -10 or greater than 4 is a solution. The open circles visually affirm that -10 and 4 are not solutions themselves.

Compound inequalities involve two separate inequalities that are combined together by "and" or "or". For absolute value inequalities, such as \(|3+x| > 7\), we often find ourselves dealing with an "or" situation due to the nature of absolute values.

The inequality \(|3+x| > 7\) leads to the compound inequality system, \(x > 4\) or \(x < -10\).

• **Or inequalities** state that **at least one** condition must be true. For \(x\) in this problem, it can be either more than 4 or less than -10.
• When solving, we treat each inequality separately and then combine results to establish a broader solution set.

The representation is crucial because many students confuse "and" and "or" compounds. Remember, "or" stands for inclusiveness in a way that we accept solutions from either inequality. For this problem, the combination \(x > 4\) or \(x < -10\) reflects in an interval union \((-\infty, -10) \cup (4, \infty)\), covering a wide range of possible values for \(x\).