91Ó°ÊÓ

Understanding the solution set is crucial when solving absolute value equations. In essence, a solution set includes all the possible values of the unknown variable that satisfy the equation. For the equation given, \(3|y+5|=12\), we want all the values of \(y\) that make this statement true. By isolating the term \(|y+5|\) on one side of the equation and solving the resulting set of possible equations, we find the complete solution set.
This solution set tells us about the values of \(y\) that the original equation can accommodate. In this exercise, the solution set comprises the values \(y = -1\) and \(y = -9\). These are the values that satisfy the equation when plugged back into the original statement, confirming their validity.

Isolating the absolute value expression is the first step in solving absolute value equations. Our goal is to have the absolute value expression by itself on one side of the equation. This is crucial because it simplifies the equation and lays the foundation for solving the rest of it.
In the equation \(3|y+5|=12\), dividing both sides by 3, isolates \(|y+5|\) as \(|y+5| = 4\). This isolation means we are now prepared to break down the absolute value into two potential equations.
Isolating is a pivotal part of simplifying challenging equations, turning them into more manageable forms that are easier to solve. Without correct isolation, it becomes far more difficult to understand and solve the equation accurately.

After isolating the absolute value expression, the next step is to tackle the secondary equations. This involves exploring the two possible conditions that the absolute value expression can satisfy.
Remember, the absolute value, by its definition, can yield either a positive or a negative result. For our isolated expression \(|y+5| = 4\), we set up two separate equations: \(y+5 = 4\) and \(y+5 = -4\).
These secondary equations represent the reality that the absolute value can describe either scenario. Solving these equations gives us the solutions \(y = -1\) and \(y = -9\). Solving secondary equations is key to understanding the nature of absolute values and to unlocking the complete set of solutions for the problem at hand.