91Ó°ÊÓ

Set notation is a way of describing a collection of objects or numbers. In mathematics, we often use it to list the solutions to an equation. The set is usually enclosed in curly braces {}, and each item within the set is separated by a comma. For example, if you solve an equation and find two solutions, 3 and 7, you would write this in set notation as {3, 7}. It’s a handy way to show all possible solutions simultaneously.
In our example, we found the solutions to the equation \(|x-2|=6\) to be \(x=8\) and \(x=-4\). Using set notation, we write this as {8, -4}. So, anytime you see an equation with multiple solutions, remember to use set notation to neatly present your answers.

Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations are of the form \(ax + b = c\), where a, b, and c are constants. The main objective is to isolate the variable on one side of the equation.
To solve it, you generally perform these steps:

• First, simplify both sides of the equation if necessary.
• Next, use addition or subtraction to get \(\textbf{all terms containing the variable}\) on one side and constant terms on the other.
• Finally, use multiplication or division to isolate the variable.

In our specific example \(x-2=6\), we started by adding 2 to both sides to isolate \(x\):

\[x - 2 + 2 = 6 + 2\]

This simplifies to:

\[x = 8\]
Similarly, for \(x-2=-6\):

\[x - 2 + 2 = -6 + 2\]

This simplifies to:

\[x = -4\]
These linear equations show the steps needed to isolate and find the variable, making sure we correctly handle both the positive and negative scenarios.

The absolute value of a number is its distance from zero on the number line, regardless of direction. It’s always a non-negative number. A key concept with absolute values is understanding that \( |a| = b \) (where b is positive) implies two possible equations: \(a = b\) and \(a = -b\).
For example, in the equation \( |x-2| = 6 \), the absolute value tells us that the expression inside the absolute value (x-2) could either be 6 or -6. This is because both 6 and -6 are 6 units away from zero. Therefore, we break it into two separate equations:

\[x - 2 = 6\]

and

\[x - 2 = -6\]
Once you solve these two linear equations, you get the possible solutions for x, considering both the positive and negative scenarios. This method ensures we cover all possible solutions to the absolute value equation.
Understanding the nature of absolute values and how they interact in equations is crucial for correctly solving and interpreting the solutions.