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To solve an absolute value equation, the first step is to isolate the absolute value expression. This involves getting the absolute value term by itself on one side of the equation.
Let's look at an example problem to understand this process better:
\(|t| + 1.1 = 6.6\).
We start by isolating the absolute value term \(|t|\). This means we need to remove any other terms on the same side as \(|t|\).
Here, we have \(+1.1\), which we can eliminate by subtracting 1.1 from both sides of the equation:
\[|t| + 1.1 - 1.1 = 6.6 - 1.1\]
This simplifies to: \[|t| = 5.5\]
Now that the absolute value term is isolated, we can proceed to solve the equation in the next step.

An absolute value equation can represent two different scenarios because the absolute value of a number is always positive.
Once the absolute value expression is isolated, we need to set up two separate equations to represent these scenarios.
Using our example \[|t| = 5.5\]:

• Scenario 1: The value inside the absolute value is positive. In this case, \[t = 5.5\].
• Scenario 2: The value inside the absolute value is negative. Here, \[t = -5.5\].

Therefore, the solutions for our equation \[|t| = 5.5\] are t=5.5 and t=-5.5.
Both of these scenarios need to be considered to completely solve the absolute value equation.

After finding the solutions, we often express them in a concise and elegant way using set notation.
Set notation is a method of listing multiple solutions within curly brackets \(\brace\).
Let's continue with our previous example:
We found two solutions \[t = 5.5\] and \[t = -5.5\].
To write these solutions using set notation, we combine them into a set:
\[t = \{5.5, -5.5\}\]
This indicates that the solution to the equation includes both values 5.5 and -5.5.
Set notation is particularly useful when dealing with multiple solutions, as it simplifies and clarifies the result.
Remember, every time you solve an absolute value equation with multiple solutions, represent your answer using set notation for clarity and precision.