91Ó°ÊÓ

Solving absolute value equations involves finding the values of the variable that make the equation true. An absolute value equation is an equation in which the expression inside the absolute value symbols is set equal to a number. Absolute value, denoted by \(\textbar x\textbar\), represents the distance of a number from zero, regardless of direction on the number line. This means that \(\textbar x \textbar \) will always be non-negative.

The general approach to solving absolute value equations follows these steps:

• Set up the equation by placing the absolute value expression equal to a given number.
• Isolate the absolute value term on one side of the equation.
• Create two separate equations: one for the positive and one for the negative scenario.
• Solve each equation separately.

Let’s apply these steps to the given exercise where \(f(x) = \textbar x \textbar - 3\) and we need to find all \(x \) for which \(f(x) = 5.7. \) This translates to solving the equation \( \textbar x \textbar - 3 = 5.7.\)

Isolating the absolute value term means getting \(\|x\|\) alone on one side of the equation. This is important because it sets up the equation for the next step where it's solved for both its positive and negative scenarios.

In the exercise, we start with the equation \(\|x\| - 3 = 5.7.\) Our goal is to isolate \(\|x\|.\)

To do this, we need to perform operations that will leave \(\|x\|\) alone on one side of the equation. Here, we can add 3 to both sides:
\[\|x\| - 3 + 3 = 5.7 + 3\] Which simplifies to:
\[\|x\|= 8.7\]
This means that the distance of \(x\) from zero is 8.7. At this point, the absolute value term is isolated, and we can move on to finding the solutions.

After isolating the absolute value term, we need to find all possible values of \(x\) that satisfy the equation. An absolute value equation \(\|x\| = a \), where \(a\) is a positive number, results in two solutions: \(x = a\) and \(x = -a\).

Referring back to our exercise, we have the equation \(\|x\| = 8.7\). This translates into two separate linear equations:
\[x = 8.7\]
\[x = -8.7\]
These solutions indicate that \(x\) can be either 8.7 or -8.7, both satisfying the original absolute value equation \(\|x\| - 3 = 5.7\).

Therefore, the solutions to the given exercise are \(\boxed{8.7 \text{ and } -8.7}.\)

Remember, whenever solving these types of problems, always isolate the absolute term first, and then consider both the positive and negative scenarios for the solutions.