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Absolute value inequalities, like \(|3x - 7| \geq 2\), can seem confusing at first. The key is to understand that the expression inside the absolute value, \(3x - 7\) in this case, can either be greater than or equal to 2, or less than or equal to -2. This gives us two separate inequalities to solve:

• \(3x - 7 \geq 2\)
• \(3x - 7 \leq -2\)

For the first inequality, add 7 to both sides to get \(3x \geq 9\). Then, divide each side by 3, leaving \(x \geq 3\).
For the second, also add 7 to both sides: \(3x \leq 5\), then divide by 3 to find \(x \leq \frac{5}{3}\).
These solutions together express the values \(x\) can take to satisfy the original inequality. The solution, therefore, is the union of these two sets: \(x \geq 3\) or \(x \leq \frac{5}{3}\). This makes solving inequalities much clearer and organized.

Once you have the solution from the inequalities, it's time to express them in interval notation. Interval notation is a shorthand used to describe the set of solutions in a compact form.For \(x \leq \frac{5}{3}\), the interval notation is \((-\infty, \frac{5}{3}]\). This means the interval extends to negative infinity and includes \(\frac{5}{3}\) (hence the square bracket).
Similarly, \(x \geq 3\) is written as \([3, \infty)\). Here, we include 3 by using a square bracket and extend to positive infinity.The solution for the absolute value inequality combines these into \((-\infty, \frac{5}{3}] \cup [3, \infty)\). The union symbol \(\cup\) indicates that any \(x\) in either interval satisfies the inequality.
This notation gives a crisp and clear way to convey the complete solution set.

Visualizing solutions on a number line helps in understanding them better. To graph the solution \((-\infty, \frac{5}{3}] \cup [3, \infty)\), follow these steps:

• For \((-\infty, \frac{5}{3}]\), shade the line from negative infinity up to and including \(\frac{5}{3}\). You'll place a closed dot at \(\frac{5}{3}\) to show it's included in the solution.
• For \([3, \infty)\), shade the line starting from 3 onwards to positive infinity. Here, also use a closed dot at 3 because it is part of the solution.

Keep these shaded areas separate, as they represent distinct intervals where the inequality holds true.
Graphing makes it easier to communicate and understand where solutions lie on a number line, providing a clear, visual representation of the problem.