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Absolute value inequalities like \(|3x + 5| < 17\) are solved by breaking them down into two separate inequalities. This involves creating a range where the expression inside the absolute value, in this case, \(3x + 5\), lies. The principle seen here is that \(|a| < b\) can be rewritten as \(-b < a < b\). This tells us not only that \(a\) is less than \(b\) but also greater than \(-b\).
So initially, we start by rewriting the given inequality:

• \(-17 < 3x + 5 < 17\)

From here, our goal is to isolate the variable \(x\) in the middle. We achieve this by simplifying each part of the inequality step-by-step. Start by subtracting 5 from all sides:

• \(-22 < 3x < 12\)

Finally, divide all sides by 3 to solve for \(x\):

• \(-\frac{22}{3} < x < 4\)

This gives you the range of values for \(x\) that satisfy the inequality.

Graphing inequalities involves visually representing the solution on a number line, which indicates where the solutions of the inequality lie. For the inequality \(-\frac{22}{3} < x < 4\), our task is to show this range.
Begin by drawing a straight horizontal line, which serves as your number line. Then identify the specific points -\(\frac{22}{3}\) and \(4\) on this line. Since we are dealing with strict inequalities (less than, not less than or equal to), use open circles or parentheses at the points.
An open circle indicates that the value at that point is not included in the solution. With these points marked, shade the region between them. This shaded portion represents all the possible values \(x\) can take according to the inequality. Ensure the shading accurately reflects the continuous set of solutions within these boundaries.

A number line offers a simple way to visually illustrate the set of solutions for inequalities, helping to comprehend them better. When representing \(-\frac{22}{3} < x < 4\), focus on these points:
Locate \(-\frac{22}{3}\) and \(4\) on the drawn number line. The key here is using open parentheses or circles to denote that these boundary points are not included. Draw such marks directly on the number line at \(-\frac{22}{3}\) and \(4\).
Following placement, draw a thick or bold line connecting the two points. This highlights the interval \(x\) falls within. Remember, the openness of endpoints signifies that the solution doesn't include the exact values at these points, just everything in between. This complete visualization aids in better understanding and communicating the solution of the inequality.