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Absolute value inequalities involve expressions within the absolute value function. The absolute value measures the distance of a number from zero on the number line. As such, an absolute value inequality like \( \left| \frac{3x - 3}{4} \right| < 6 \) suggests examining how far the expression inside the absolute value is from zero.

To solve such inequalities, you represent the absolute value expression as two separate inequalities without the absolute value sign. For \( \left| A \right| < B \), it translates to \( -B < A < B \). So, our expression \( \left| \frac{3x - 3}{4} \right| < 6 \) becomes two inequalities:

• \( -6 < \frac{3x - 3}{4} \)
• \( \frac{3x - 3}{4} < 6 \)

By dealing with the expression inside the absolute value, you capture all points \(x\) so that the left and right distances from zero are less than 6. This helps you determine the range of possible solutions.

Graphing solutions on a number line visually demonstrates the range of values that satisfy an inequality. For expressions like \( -7 < x < 9 \), it's helpful to portray this visually.

First, a number line is drawn, marking crucial boundary points such as -7 and 9. Using open circles to mark these points indicates that these endpoints are excluded from the solution set, reflecting the 'less than' nature of our inequality— "not equal to."

Between these points, shade the region to show that all values of \(x\) within this interval satisfy the inequality. This graphical representation is a quick and clear way to show or verify where your solution lies. It helps ensure no steps were missed during solving, giving a clear visual confirmation.

To solve inequalities effectively, algebraic manipulation is key. This involves carefully executing operations to isolate the variable \(x\) while maintaining the inequality's integrity.

Start with breaking the absolute value inequality into its two components, as discussed. For each part of the inequality, execute identical steps to keep the process balanced:

• Multiply each side by 4 to clear the fraction. This gives the simplified forms: \(-24 < 3x - 3\) and \(3x - 3 < 24\).
• Add 3 to every term, which results in: \(-21 < 3x\) and \(3x < 27\).
• Finally, divide every term by 3 to completely isolate \(x\), achieving \(-7 < x\) and \(x < 9\).

Each algebraic step builds upon the last, guided by operations that maintain equality or inequality. Care is taken to apply these consistently to every part of the inequality. This ensures the solution is both correct and comprehensive.