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Interval notation is a handy mathematical expression used to describe a set of numbers, typically solutions to inequalities. Instead of listing all numbers in the set, interval notation succinctly provides the range of possible values. Here's how it works:

Generally, brackets or parentheses are used to denote the endpoints of the intervals:

• A square bracket [ ] indicates that the endpoint is included in the interval, known as inclusive.
• A parenthesis ( ) implies the endpoint is not included, called exclusive.

If a solution spans multiple intervals, we use the union symbol \(\cup\) to combine them. For example, in our solution to the absolute value inequality, we use

• \((-\infty, -3]\)
• \([-1, \infty)\)

Here, \((-\infty, -3]\) includes all numbers less than or equal to -3, and \([-1, \infty)\) includes all numbers greater than or equal to -1. The union \(\cup\) signifies that numbers in either interval satisfy the original inequality.

Solving absolute value inequalities involves a few specific steps that allow us to find the range of values a variable can take. Let's break it down out using the example inequality \(|2x + 4| \geq 2\):

First, we need to understand the meaning of the absolute value. When \(|2x + 4| \geq 2\), it indicates that the expression \(2x + 4\) can be either:

• Greater than or equal to 2 (\(2x + 4 \geq 2\))
• Or less than or equal to -2 (\(2x + 4 \leq -2\))

This dual scenario arises because the absolute value either retains a positive value or negates a negative to positive.

Once these scenarios are set, we solve each inequality separately:

In case of \(2x + 4 \geq 2\), subtract 4 from both sides to get \(2x \geq -2\), then divide by 2, resulting in \(x \geq -1\). For \(2x + 4 \leq -2\), subtract 4 to get \(2x \leq -6\) and divide by 2, yielding \(x \leq -3\).

The solutions \(x \geq -1\) and \(x \leq -3\) are separate due to the nature of the original inequality.

Graphing solution sets of inequalities helps visualize the solution on a number line, making it easier to comprehend. When we graph the solution set for \((-\infty, -3] \cup [-1, \infty)\), we use the following steps:

First, draw a horizontal line to represent the number line. Place markers on it at \(-3\) and \(-1\). Since the intervals are inclusive at these points, draw filled circles at these values:

• A filled circle at \(-3\) signifies that -3 is part of the solution (\([, ]\))
• A filled circle at \(-1\) confirms -1's inclusion as well

Next, shade the number line to the left of \(-3\) towards negative infinity, and to the right of \(-1\) towards positive infinity.
These shaded regions represent solutions satisfying \(|2x + 4| \geq 2\). They visually show numbers belonging to either condition, as illustrated by the union of intervals.