Problem 72 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ \left|\frac{2}{5}\right| \quad\left|-\frac{2}{5}\right| $$

Step-by-Step Solution

Answer

The correct symbol is \( = \).

Step 1: Identify Absolute Values

To solve this exercise, we first need to identify the absolute values given. We have two expressions: \( \left| \frac{2}{5} \right| \) and \( \left| -\frac{2}{5} \right| \). The absolute value of a number is always non-negative, regardless of whether the number inside is positive or negative.

Step 2: Calculate Each Absolute Value

Next, we calculate the absolute values for each number. For \( \left| \frac{2}{5} \right| \), the absolute value is \( \frac{2}{5} \). For \( \left| -\frac{2}{5} \right| \), the absolute value is also \( \frac{2}{5} \). This is because the absolute value function removes the negative sign.

Step 3: Compare the Absolute Values

Now that we have both absolute values as \( \frac{2}{5} \), we compare them: \( \frac{2}{5} \) and \( \frac{2}{5} \). Since both values are identical, the appropriate comparison symbol between them is \( = \).

Key Concepts

Inequalities

Inequalities refer to mathematical expressions that involve symbols like \(<\), \(>\), and \(=\). These symbols are used to compare two values and show how they relate to each other.
- **Less than** \((<)\) indicates that one value is smaller than another.- **Greater than** \((>)\) shows that one value is larger.- **Equals** \((=)\) means that both values are the same.
In our exercise, we want to find the right comparison between the absolute values of \(\left| \frac{2}{5} \right|\) and \(\left| -\frac{2}{5} \right|\). Since both calculations result in the same value, the inequality symbol that fits is \(=\), indicating both sides are equal.

Comparison

Comparison in mathematics is about understanding the relationship between numbers or expressions. It tells us if one number is greater, smaller, or equal to another in value.
**Steps in Comparison:**

• Identify the values you want to compare. In this case, we have the absolute values \(\left| \frac{2}{5} \right|\) and \(\left| -\frac{2}{5} \right|\).
• Calculate or evaluate each value to see what they are numerically. Here, both values are calculated to be \(\frac{2}{5}\).
• Use the correct inequality symbol based on their numerical comparison, which is \(=\) in this case.

Using comparison helps you visualize the size and order of numbers, making it easier to solve inequalities or similar mathematical problems.

Non-negative Numbers

Non-negative numbers are numbers that are either positive or zero. They do not include any negative values. The concept of non-negative numbers is essential when working with absolute values.
**Absolute Value and Non-Negativity:**

• The absolute value of any real number is always non-negative. It measures the distance of a number from zero on a number line.
• Whether a number is positive or negative, its absolute value will transform it into a non-negative number. For example, both \(\left| \frac{2}{5} \right| \) and \(\left| -\frac{2}{5} \right| \) give \(\frac{2}{5}\), a non-negative outcome.
• This property is crucial when comparing or solving problems involving absolute values and inequalities.

Understanding non-negative numbers ensures clarity when dealing with expressions that can include negative integers, especially when extracted through operations such as taking absolute values.