Problem 86 - Elementary Algebra

Question

If \(a<0,\) then \(|a|=\) ____________.

Step-by-Step Solution

Answer

Answer: If a < 0, then the absolute value |a| = -a.

Step 1: Understand the absolute value function

The absolute value function, represented by |x|, equals x if x is positive or zero, and equals -x if x is negative. In other words, if x ≥ 0, then |x| = x; if x < 0, then |x| = -x.

Step 2: Apply the absolute value function to the given number 'a'

Since 'a' is negative (a < 0), according to the absolute value function, we need to find |-a|. This means finding the non-negative value of 'a'.

Step 3: Find the absolute value of 'a'

Since 'a' is negative, to find the absolute value, we need to negate 'a' again. Thus, |a| = -(-a) which equals 'a'. So, if a < 0, then |a| = -a.

Key Concepts

Negative Numbers

Negative numbers are an important part of the number line and help us understand quantities that are less than zero. They are represented by a minus sign in front of a number, like -3 or -7. In practical situations:

• Negative numbers can represent debts, such as owing $5 would be denoted as -5.
• They can also depict temperatures below freezing, like -10°C.

Negative numbers are crucial in mathematics for solving problems that involve subtraction that crosses zero, such as when the result of a subtraction is less than the number being subtracted. Understanding how negative numbers work, including how they behave under different operations, is key to gaining a deeper insight into math.

Mathematical Functions

Mathematical functions are rules that assign each input exactly one output. These functions help us describe relationships and changes between quantities. In the realm of absolute value:

• The absolute value function is a type of mathematical function that maps any real number to its distance from zero on the number line.
• This means the output of the absolute value function is always non-negative.

For example, for a given number, if it is negative, the absolute value function 'flips' it to a positive number. On the other hand, if the number is already positive or zero, the absolute value leaves it unchanged. This is why the absolute value function is important, as it simplifies various mathematical operations involving negative numbers.

Elementary Algebra

Elementary algebra deals with the basics of algebra, focusing on understanding mathematical symbols and their relationships. A primary concept in algebra is the manipulation and simplification of expressions. This includes handling functions like absolute value:

• The basic operation involves converting expressions to a standard form that can be easily understood and utilized.
• For example, a basic task in elementary algebra could be evaluating \( |x| \) for a specific value of x.

By manipulating algebraic expressions, students learn to solve equations and understand mathematical relationships better. Understanding how to transform and solve expressions involving absolute values is a foundational skill in elementary algebra, enabling students to handle more complex algebraic problems in the future.