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The absolute value of a number refers to its distance from zero on the number line, without considering its direction. For instance, both \(-7\) and \(7\) have an absolute value of \(7\) because they are both seven units away from zero. Absolute values are always non-negative.
However, it's essential to consider additional signs outside the absolute value when evaluating expressions. In the case of \(|-6|\), the absolute value is simple \(6\), but if we have \(-|6|\), it becomes \(-6\) due to the negative sign placed in front of the absolute value.
This can initially seem puzzling but remembering to revert to the positive form first, before applying any additional operations, simplifies the process.

Evaluating expressions involves simplifying them to their most basic form. This could include operations like addition, subtraction, and multiplication.
In our initial problem, evaluating involves calculating absolute values and handling negative signs. For instance, \(-(-9)\) simplifies to a positive \(9\) as the two negatives cancel each other out, and \(-4\) remains \(-4\).
These preliminary steps are crucial in constructing a clear path towards the correct arrangement of numbers. By breaking each term down to its simplest form, it becomes easier to compare and order them appropriately.

Negative numbers are values less than zero and typically written with a negative sign \((-\)). They represent directions opposite to positive numbers on the number line.
In the given exercise, numbers like \(-4\), \(-7\), and \(-6\) are negative. The placement of the negative sign in front of these numbers offers a crucial difference, as it indicates their opposite nature in comparison to their absolute masses.
It's necessary to treat negative numbers with care, especially in operations and comparisons, as they behave differently than positive numbers. For example, \(-7\) is smaller than \(-4\), despite seven being numerically larger than four.

A number line is a visual representation of numbers placed sequentially according to their value. It helps in understanding the order of numbers, positive and negative, straightforwardly.
Zero is positioned at the center of the number line, with positive numbers increasing towards the right, and negative numbers decreasing towards the left. This setup assists significantly in arranging numbers as it visually differentiates between their magnitudes.
In our problem, plotting \(-7\), \(-6\), \(-4\), and \(+9\) on a number line allows for an intuitive interpretation of their order from left (smallest) to right (largest). Such visualization highlights the relative positions and helps refine understanding of numerical concepts.