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Understanding how to graph inequalities is crucial when visualizing the solution sets of these expressions. In the exercise provided, after determining that the value of y is less than or equal to -6, it's important to depict this on a number line.

Start by drawing a horizontal line with numbers placed at equal intervals. Locate -6 on this line. Since the inequality is \( y \leq -6 \) (meaning y is less than or equal to -6), you need to draw a solid circle on -6 to indicate that -6 itself is included in the solution set. Then, draw a shaded line or arrow extending to the left of -6 to represent all the numbers less than -6 that are also part of the solution.

Isolation of variables is a fundamental step in solving inequalities. It involves manipulating the inequality to get the variable by itself on one side, so you can clearly see the solutions or range of solutions. In the exercise, we began by subtracting 14 from both sides to isolate the -4y term.

It's crucial to perform the same operation on both sides to maintain the balance of the inequality. Think of the inequality as a scale that must remain equal on both sides. Once you isolate the variable, you get a clearer picture of what values the variable can take. In our case, isolating -4y resulted in \( -4y \geq 24 \).

One of the more nuanced concepts when solving inequalities is dividing both sides by a negative number. In such situations, the direction of the inequality sign must be reversed. As seen in the exercise, dividing \( -4y \geq 24 \) by -4 to solve for y requires flipping the sign to maintain the true relationship between the values.

This is because, in mathematics, the act of dividing by a negative number is essentially 'flipping' all the values across zero on the number line. If we didn't reverse the sign, the inequality would incorrectly suggest that higher values of y were solutions when it fact it is the lower ones. Therefore, the correct result after division is \( y \leq -6 \), which conforms with this reversal rule when dividing by negative numbers.