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Interval notation is a way to describe a set of numbers along a number line. It's compact and easy to read. This notation helps us express solution sets of inequalities in a clear manner. For absolute value inequalities like \( |3x - 2| \geq 5 \), once we solve it, the solution is given in interval notation.

In this case, we found two solutions:

• \( x \geq \frac{7}{3} \)
• \( x \leq -1 \)

These correspond to two intervals:

• \( [\frac{7}{3}, \infty) \)
• \( (-\infty, -1] \)

We combine these intervals using the union symbol \((\cup)\), resulting in the complete solution \((-\infty, -1] \cup [\frac{7}{3}, \infty)\). The use of \([\text{ and }]\) brackets shows whether an endpoint is included or not.

A square bracket, \([\text{ or }]\), means the endpoint is included in the interval (known as 'closed'). A parenthesis, \((\text{ or })\), means the endpoint is not included (known as 'open'). In our solution, both intervals \([-1] \text{ and } [\frac{7}{3}]\) are closed because they are the exact points where the inequality holds true.

Graphing inequalities on a number line gives a visual representation of the solution set. This makes it easier to see which ranges are included in the solution. For the inequality \(|3x - 2| \geq 5\), we determined the intervals are\((-\infty, -1] \cup [\frac{7}{3}, \infty)\).

To graph these:

• Draw a number line.
• Mark the critical points \(-1\) and \(\frac{7}{3}\).
• Use a closed circle at both \(-1\) and \(\frac{7}{3}\) because these points are included in the solution.
• Shade the number line to the left of \(-1\) and to the right of \(\frac{7}{3}\).

These shaded regions indicate that all the numbers in these parts of the number line satisfy the original inequality. Graphing is a useful tool because it complements interval notation and allows one to quickly verify which values are solutions visually.

Solving inequalities with absolute values might seem complex, but when broken down, they follow simple steps. For \(|3x - 2| \geq 5\), the process involves:

Step 1: Remove the absolute value.

• Separate into two inequalities.
• \(3x - 2 \geq 5\) and \(3x - 2 \leq -5\).

Step 2: Solve each inequality.

• For \(3x - 2 \geq 5\):
• Add 2 to both sides to get \(3x \geq 7\).
• Divide by 3 to isolate \(x\), giving \(x \geq \frac{7}{3}\).
• For \(3x - 2 \leq -5\):
• Add 2 to both sides, resulting in \(3x \leq -3\).
• Divide by 3 to get \(x \leq -1\).

Step 3: Combine the solutions.

• Realize solutions are \(x \geq \frac{7}{3}\) or \(x \leq -1\).
• Use this information to express in interval notation: \((-\infty, -1] \cup [\frac{7}{3}, \infty)\).

Breaking down each step clarifies what each part of the inequality represents and ensures we understand how to reach the solution systematically. As you follow each phase, solving these inequalities becomes straightforward.