Problem 48 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lesson 2.2) $$-|5|$$

Step-by-Step Solution

Answer

-5.

Step 1: Evaluate the Absolute Value

First, calculate the absolute value of 5. The absolute value of any number, whether positive or negative, is always non-negative. In the case of 5, the absolute value is simply 5. Therefore, we can rewrite \(-|5|\) as \(-5\).

Step 2: Evaluate the Negation

The result of Step 1 is -5. Since there is no mathematical operation left, the resultant value stays -5.

Key Concepts

Absolute Value

The absolute value of a number is a basic yet pivotal concept in mathematics, representing the distance of that number from zero on the number line, regardless of direction. In simpler terms, it is the non-negative value of the number without considering its sign. For any real number 'x', the absolute value is denoted by |For example, the absolute value of both -3 and 3 is 3. When you come across an expression like |-5|, you should interpret it as the distance 5 units away from zero, which is numerically 5.
To clarify, when evaluating the expression , one would first address the absolute value of 5, which simplifies to 5, since it is already a positive number. Understanding absolute values is essential for problem-solving, especially when dealing with real-world scenarios where negative values may not be applicable or when performing operations like subtraction that could result in positive or negative differences.

Negation

In mathematics, negation refers to the operation of changing the sign of a number, effectively reflecting it across the zero point on the number line. When you negate a positive number, it becomes negative, and vice versa. The operation is symbolized by a minus sign (-) placed before the number.
For instance, the negation of 7 is -7, and similarly, the negation of -7 is 7. In the context of the exercise , after we've identified that the absolute value of 5 is 5, we move to the negation step which involves simply applying a negative sign to our positive 5 resulting in -5. This is the essence of negation—altering the positivity or negativity of a value without changing its magnitude. Students should grasp that negation does not affect the 'size' of the number, but its 'direction' on the number line.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, division, and exponentiation) without an equal sign. They are fundamental in representing relationships in algebra and solving various mathematical problems.
An expression like involves an absolute value and negation, both of which are common operations in algebra. To evaluate such expressions, one must follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our example, since there are no parentheses, exponents, multiplication, or division, we move straight to addressing the absolute value and then apply the negation. Fully understanding algebraic expressions and their components, like absolute value and negation, is key to succeeding in algebra and further math courses.