91Ó°ÊÓ

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Variables represent unknown values and can stand in for any number. Operations can include addition, subtraction, multiplication, and division, and more complex functions. Understanding algebraic expressions is fundamental in algebra, as they are used to represent real-world relationships and to solve various mathematical problems. For example, the expression \(|x+y|\) involves two variables, where 'x' and 'y' could be any numbers. The vertical bars denoting the absolute value imply another layer of operation on the expression within them.

To evaluate algebraic expressions like \(|x+y|\) for specific values, we employ substitution, which involves replacing the variables with the numbers they represent, and then proceeding with the usual order of operations to simplify and solve the expression.

The substitution method is a technique used in algebra to find the value of an expression by replacing the variables with actual numbers. This method helps in simplifying the problem into a more manageable form. To use this method effectively, you simply replace each variable in the expression with the value it represents.

In the context of the original exercise, we substituted the values of 'x' and 'y' into the algebraic expression \(|x+y|\) to get \(|2+(-5)|\). This substitution simplifies our problem, making it easier to understand and solve. It's crucial to follow the proper order of operations even after substitution, first handling the arithmetic inside the absolute value brackets before proceeding to deal with the absolute value itself.

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. Absolute value properties are important to understand because they're used in a variety of contexts within mathematics, such as solving equations and inequalities.

Some key properties of the absolute value include: \(|-a| = |a|\), which asserts that the absolute value of a number is always positive no matter if the original number is negative or positive, and \(|ab| = |a||b|\), which states that the absolute value of a product is the product of the absolute values. Another property is the triangle inequality, which algebraically can be represented as \(|a+b| \text{ is always }\ \text{less than or equal to }\ |a|+|b|\). When evaluating the absolute value of an expression, after substituting the numbers and simplifying, we apply these properties to find the final result. For instance, in our exercise, after substituting and simplifying, we needed to evaluate the absolute value of -3, which is 3 based on the definition that it is the distance from zero, disregarding the sign.