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Interval notation is a shorthand used to describe ranges of numbers that satisfy an inequality. Unlike inequality notation, which uses symbols such as '<' and '>', interval notation uses brackets and parentheses to denote intervals on the real number line. For instance, the solution to the inequality \( |x| < 7 \) was found to be \( -7 < x < 7 \). In interval notation, this is represented as \( (-7, 7) \).

Here is how you interpret interval notation:

• Open parentheses, \( ( \), indicate that the endpoint is not included in the interval.
• Closed brackets, \( [ \), signify that the endpoint is included.

The notation \( (-7, 7) \) tells us x can be any number greater than -7 but less than 7, without including -7 and 7 themselves. This method of notation is particularly helpful when graphing inequalities or expressing solutions in a concise manner.

Solving inequalities is a fundamental skill in mathematics that involves finding all values that satisfy a given inequality. The process is similar to solving equations but requires the extra step of considering the direction of the inequality sign when multiplying or dividing by a negative number.

For the absolute value inequality \( |x| < 7 \), the steps are:

1. Split the inequality into two cases, considering both the positive and negative scenarios.
2. Solve each resulting inequality, remembering to reverse the inequality sign when multiplying by negative one.
3. Combine the solutions to represent a range of values that satisfy the original inequality.

Remember, the critical element is to flip the inequality sign whenever you multiply or divide by a negative number to maintain the true relationship between the values.

The absolute value of a number measures its distance from zero on the number line, regardless of direction. Symbolically, it is represented by two vertical bars, e.g., \( |x| \). It always results in a non-negative value because distance cannot be negative. For example, \( |-3| = 3 \) and \( |3| = 3 \).

In the case of inequalities, the absolute value creates two scenarios: one where the variable within the absolute value is positive, and one where it is negative. Therefore, the inequality \( |x| < 7 \) tells us that the distance of x from zero has to be less than 7, which is why we split into \( x < 7 \) and \( -x < 7 \) to find all values of x within that distance from zero. Understanding the concept of absolute value is crucial for solving absolute value inequalities effectively.