Problem 72 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ \left|\frac{2}{5}\right| \quad\left|-\frac{2}{5}\right| $$

Step-by-Step Solution

Answer

\(=\)

Step 1: Identify Absolute Values

First, identify the absolute value of each number. The absolute value of a number is its distance from zero on the number line without considering direction. Hence, irrespective of the sign, the absolute value of \(\frac{2}{5}\) and \(-\frac{2}{5}\) is \(\frac{2}{5}\).

Step 2: Comparing Absolute Values

Since the absolute values of both \(\frac{2}{5}\) and \(-\frac{2}{5}\) are the same, i.e., \(\frac{2}{5}\), the correct relation between them is an equality.

Key Concepts

compare fractions

When comparing fractions, it is essential to understand the value each fraction represents relative to others. The fraction \(\frac{2}{5}\) can be compared to another fraction, such as \(-\frac{2}{5}\), by considering their absolute values. Absolute value helps simplify the comparison by focusing only on the size of the fractions, disregarding whether they are positive or negative. This means you are comparing how far they are from zero, not in which direction they lie from zero. Let's break it down:

• The numerator (top number) tells us how many parts we have.
• The denominator (bottom number) tells us into how many parts the whole is divided.

When the numerators are positive, a larger numerator means the fraction is larger if denominators are equal. Here, both \(\frac{2}{5}\) and \(-\frac{2}{5}\) have the same numerators and denominators, so their sizes are equal when considering absolute values.

number line

The number line is a fundamental tool in mathematics for visually representing numbers and understanding their relationships, especially distance and order. It's a straight line where numbers are placed at equal intervals. Positive numbers are to the right of zero, and negative numbers are to the left.Using a number line helps to

• Clearly see the position of fractions like \(\frac{2}{5}\) and \(-\frac{2}{5}\).
• Understand that every number on the number line has a corresponding absolute value, which is its distance from zero.

To find absolute values on the number line:

• Identify the position of a fraction without considering its sign.
• Count the number of spaces from zero to this position; this is its absolute value.

Thus, both \(\frac{2}{5}\) and \(-\frac{2}{5}\) have equal absolute value because they are equally spaced from zero.

distance from zero

Distance from zero aka absolute value, tells you how far a number is from zero, ignoring its direction. It's a concept closely tied to absolute values.Here's why it matters:

• Absolute value makes comparing numbers with different signs straightforward.
• Understanding distance from zero on a number line clarifies inequalities and equalities.

The equation for absolute value is simple: \(|a| = a\) if \(a\) is positive and \(|a| = -a\) if \(a\) is negative. For example, \(|-\frac{2}{5}|\) is \(\frac{2}{5}\).When comparing \(|\frac{2}{5}|\) and \(|-\frac{2}{5}|\), both are the same because both are the same distance from zero on the number line. Therefore, understanding absolute values as distances helps appreciate why both these fractions are equal in size when considering their absolute values.