91Ó°ÊÓ

Absolute value refers to the distance of a number from zero on the number line, without considering which direction from zero the number lies. This means absolute values are always non-negative. For example, the absolute values of both -3 and 3 are 3, written as \(|-3| = 3\) and \(|3| = 3\).\
\
There are crucial properties of absolute values when solving equations. One important property is that \(|a| = |b| = a = b \text{ or } a = -b|\). So the equation \(|x-2| = |x+2|\) implies two possibilities: either \(x-2 = x+2\) or \(x-2 = -(x+2)\).\
\
Understanding absolute values can simplify the approach to solving equations that compare absolute values or use them in different algebraic contexts.

Set notation is a mathematical method used to describe a collection of objects, particularly numbers. In this context, the solution to an absolute value equation can be expressed using set notation to clearly indicate the numbers that satisfy the equation.\
\
Consider that any solution of an equation can often be written in set notation. For instance, if the solution is all real numbers, it can be written as \( \textbf{Set of all real numbers}\, \mathbb{R}\). This notation helps in understanding the complete range of possible solutions succinctly.\
\
In the context of our problem \(|y-2| = |2-y|\), we found that any value of y satisfies the equation. Thus, the set notation to represent all solutions would be \( \mathbb{R} \text{ (Set of all real numbers) }\).

Solving equations generally involves finding the variable values that satisfy the given equality. Here's a step-by-step approach for solving the equation \(|y-2| = |2-y|\):\
- First, recognize the absolute value property. In this case, \( |y-2| == |2-y| \) always holds true, because \( y-2 \) and \( 2-y \) are negatives of each other.\
- Next, simplify and realize that this particular equation holds for any real number y.\
- Then, express the solution using set notation to convey all possible solutions clearly.\
\
It's also important to verify the solutions by substituting back into the original equation to ensure no mistake was made. For \(|y-2| = |2-y|\), we see that indeed any real number y makes the equation true, confirming our solution.\
\
This method ensures clarity and accuracy when solving absolute value equations.