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The number line is a powerful visual tool that helps us understand numbers and their relationships. Imagine a straight line with zero in the center. Positive numbers are on the right, while negative numbers are on the left. Each step away from zero represents an increase or decrease in value.

Absolute value is closely tied to the number line. It measures how far a number is from zero, without considering direction. For example, the distance of -5 from zero is the same as the distance of 5, which is why \(|-5| = 5\). This concept shows us that absolute value is always non-negative, as it represents a distance.

When evaluating an expression like \(2|-5|\), understanding the number line helps clarify why the absolute value of -5 is 5.

Multiplication is a fundamental operation in mathematics that allows us to find the total of adding a number repeatedly. If you multiply 2 by 5, it's like adding 5 two times, which gives you 10.

When we substitute the absolute value in the expression \(2|-5|\) to get \(2 \times 5\), we're applying this repeated addition. It's critical to remember that multiplication follows specific rules, such as the commutative property, where the order of numbers doesn't matter, i.e., \(a \times b = b \times a\).

This concept simplifies calculations and helps solve problems efficiently by converting repeated additions into a more compact form.

Addition is one of the first arithmetic operations we learn. It combines two or more numbers to find their total. In our exercise, the expression started as \(3 + 2|-5|\). By substituting the multiplication result, it becomes \(3 + 10\).

When adding, the order of the numbers doesn't affect the sum due to the commutative property of addition, \(a + b = b + a\). Knowing properties like this can make problem-solving more intuitive.

Addition is also associative, which means numbers can be grouped in any way without changing the sum, i.e., \((a + b) + c = a + (b + c)\). Understanding these properties ensures that you can manipulate and simplify expressions correctly.