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Interval notation is a way to describe the solution sets of inequalities using intervals. It’s a concise method to denote all numbers that satisfy the inequality in question. When dealing with absolute value inequalities, like \(|8x + 3| > 12\), you often end up with two solution intervals because absolute value represents a distance from zero.

In the provided solution, solving \(|8x + 3| > 12\) gives us two separate conditions: \(x < -\frac{15}{8}\) or \(x > \frac{9}{8}\). These conditions tell us where the expression \(8x + 3\) is either less than -12 or greater than 12. Therefore, the solution is split into intervals:

• From negative infinity to \(-\frac{15}{8}\) (\(-\infty, -\frac{15}{8})\)
• From \(\frac{9}{8}\) to positive infinity (\(\frac{9}{8}, \infty)\)

These intervals are represented in interval notation because it's efficient and clear for anyone reviewing your work. It informs us that any number less than \(-\frac{15}{8}\) or more than \(\frac{9}{8}\) satisfies the absolute value inequality.

Inequalities are mathematical statements showing the relationship between quantities that are not necessarily equal. They often involve symbols like >, <, ≥, and ≤. In the context of absolute value inequalities, they are used to express the range of values that satisfy a condition like \(|8x + 3| > 12\).

When breaking down absolute value inequalities, you usually convert the problem into two separate inequalities. This is because the absolute value essentially measures how far a number is from zero in either direction. For instance, solving \(|8x + 3| > 12\) translates to needing to check both where \(8x + 3\) is greater than 12 or less than -12.

• The inequality \(8x + 3 > 12\) is solved for \(x\) by isolating \(x\): first subtract 3, then divide by 8 to find \(x > \frac{9}{8}\).
• Similarly, solve \(8x + 3 < -12\) to get \(x < -\frac{15}{8}\).

Once these inequalities are solved, they are combined (using 'or' for absolute value inequalities) to give the full set of solutions as x being less than \(-\frac{15}{8}\) or greater than \(\frac{9}{8}\).

Graphing solutions is a visual representation of the solution set for inequalities on a number line. It helps to quickly interpret and understand the range of solutions. After solving inequalities, especially involving absolute values like \(|8x + 3| > 12\), graphing the solution can make it easier to visualize what's happening.

To graph the solution:

• Draw a horizontal line to represent a number line.
• Identify the critical points of \(-\frac{15}{8}\) and \(\frac{9}{8}\) from the solution \[x < -\frac{15}{8} \text{ or } x > \frac{9}{8}\]
• Place an open circle on both \(-\frac{15}{8}\) and \(\frac{9}{8}\) to indicate these points are not included in the solution (since our inequality is a strict 'greater than').
• Shade the region to the left of \(-\frac{15}{8}\) indicating numbers less than \(-\frac{15}{8}\), and the region to the right of \(\frac{9}{8}\) to show numbers greater than \(\frac{9}{8}\).

Graphing not only helps cement an understanding of the solution but also provides clarity to others who view the solution. It's highly recommended for public presentations or checking work accuracy.