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Understanding the order of operations is crucial to simplifying expressions and solving algebraic equations accurately. This hierarchy, often abbreviated as PEMDAS, guides us through which steps to perform first to get to a solution. The acronym stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let's apply this order to our example: The expression \( |6+3\times(-4)| \) requires us to address the multiplication before addition. By multiplying 3 by -4, we get -12. Following this, we add 6 to our result, which gives us -6 inside the absolute value. The correct application of the order of operations ensures we simplify expressions step by step to avoid mistakes and achieve the right answer.

The absolute value of a number is its distance from zero on a number line, regardless of direction. Symbolically represented by two vertical bars \(|x|\), the absolute value function transforms any negative number into its positive counterpart. Positive numbers and zero, however, remain the same.

When simplifying \( |6+3\times(-4)| \), after applying the order of operations, we are left with the expression \( |-6| \). Here, the absolute value function comes into play, converting -6 into 6. Essentially, the absolute value nullifies any negative sign, reflecting the idea that distance cannot be negative. By grasping this concept, students can confidently work with situations that involve distance, regardless of direction.

Basic algebra is a fundamental part of mathematics that deals with symbols and the rules for manipulating these symbols. It's a building block for more advanced math and helps to describe relationships where variables are involved.

For our given problem, algebra comes into play when we replace the multiplication and addition with actual numbers. In \( |6+3\times(-4)| \), the numbers and operations work together to create a numerical expression that simplifies down to a single value. By mastering basic algebra concepts such as combining like terms, understanding variables, and simplifying expressions, students are equipped to tackle a wide range of mathematical problems and theoretical concepts.