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An expression is a mathematical phrase that combines numbers, variables, and operations. It's similar to a sentence in language that we can interpret and solve. For example, in the given problem, we have the expression \(|x-3|\). Here, \(x-3\) is the expression inside the absolute value bars, which tells us to focus on what happens in that small part of the mathematical world. This expression can include things like:

• Numbers (e.g., 1, -5, or any decimal)
• Variables, which are symbols that stand for unknown values, like \(x\)
• Operations like addition (+), subtraction (-), multiplication (*), and division (/)

Expressions can look very different, ranging from simple to complex, but they all serve the purpose of representing quantities and their relationships in a form we can work with and analyze.
In our case, understanding the expression \(x-3\) is the first step to knowing what the absolute value bars will affect.

Understanding negative numbers is crucial in solving expressions like \(|x-3|\) where \(x < 3\). A negative number is a number less than zero, and it is represented with a minus sign (-). It indicates an opposite direction or value.
When \(x < 3\), the result of \(x-3\) is negative since any number less than 3 subtracted from 3 results in a negative. For instance:

• If \(x = 2\), then \(x - 3 = 2 - 3 = -1\).
• If \(x = 0\), then \(x - 3 = 0 - 3 = -3\).

Knowing that \(x-3\) is negative lets us determine how absolute values transform the expression. When the absolute value operation is applied, it flips the negative to positive by taking the opposite, leading us to change from \(x-3\) to \(-(x-3)\). This is a fundamental property we use to simplify expressions involving absolute values.

Simplifying expressions is about finding an equivalent expression that is easier to work with. Taking our example, we started with \(|x-3|\), knowing \(x < 3\).
Once we identified that \(x-3\) is negative, we converted it to \(-(x-3)\) so that the resulting value is positive, maintaining the logic of absolute values.

• The negative sign outside the parentheses acts on everything inside, turning \(x-3\) into \(-x + 3\).
• This shows how reversing the operation inside the absolute value converts and simplifies the expression.

Remember, simplification is crucial for making expressions more manageable and ready for further math work. Whether you're solving equations or plugging numbers into an expression, clarity and simplicity make calculations easier and less error-prone.