Problem 14 - Algebra 1

Question

Evaluate the expression. $$\left|-\frac{1}{5}\right|$$

Step-by-Step Solution

Answer

The absolute value of \(-\frac{1}{5}\) is \(\frac{1}{5}\).

Step 1: Identify the negative fraction inside the absolute value

In the expression \(\left|-\frac{1}{5}\right|\), we identify that the number inside the absolute value is -1/5.

Step 2: Apply absolute value rule

The rule of absolute value states that the absolute value of a number is its distance from zero. As we're dealing with a negative fraction, its absolute value will be the corresponding positive fraction, i.e., \(\frac{1}{5}\).

Step 3: Write down the evaluated expression

After applying the absolute value rule, the expression simplifies to \(\frac{1}{5}\).

Key Concepts

Evaluating Expressions

Understanding how to evaluate expressions, especially when they include absolute values, is essential in mathematics. To evaluate an expression means to find its numerical value. When an expression contains variables, you would usually substitute these with given numbers and perform the arithmetic operations according to the order of operations: parentheses, exponents, multiplication and division, and finally addition and subtraction (PEMDAS).

In the case of our exercise \(\left| -\frac{1}{5} \right|\), there are no variables. Instead, we are asked to determine the numerical value when the absolute value is applied to a negative fraction. This brings us to an understanding of the absolute value concept and how it changes an expression.

Negative Fractions

Let's take a closer look at negative fractions, crucial elements in our exercise. A fraction represents a part of a whole, and it consists of a numerator and a denominator. When the fraction is negative, it means that we have a negative part of the whole.

Coping with negative fractions requires us to understand that they behave just like any other number when performing operations: adding, subtracting, multiplying, or dividing. However, when they fall within an absolute value, something interesting happens; their inherent 'negativeness' is stripped away. The absolute value of any number, including a negative fraction like \( -\frac{1}{5} \), will always be non-negative because absolute value measures distance on the number line, not direction.

Absolute Value Rule

The absolute value of a number reflects its distance from zero on a number line, disregarding any sign it might carry. This means that the absolute value rule helps us convert any negative number into its positive counterpart, effectively 'taking' the number's magnitude without considering its direction (positive or negative).

In our solved exercise, applying the absolute value rule to the negative fraction \( -\frac{1}{5} \) turns it into a positive \( \frac{1}{5} \), which represents the same distance from zero but in a positive direction. Remember, the absolute value of zero is zero since it is not at any distance from itself, and the absolute value of any positive number is the number itself. Hence, absolute value is often visualized as 'removing' the negative sign from negative numbers, which is especially helpful when evaluating expressions that involve negative fractions within absolute values.