Problem 63 - Algebra A Combined Function

Question

Place in the appropriate space to make each a true statement. $$ |-5| \quad-(-5) $$

Step-by-Step Solution

Answer

Place '=' between the expressions: 5 = 5.

Step 1: Understand Absolute Value

Absolute value represents the distance of a number from zero on the number line. For any number \(x\), \(|x|\) is always non-negative. In our case, \(|-5| = 5\), because the distance from \(-5\) to 0 is 5.

Step 2: Simplify the Negative of a Negative

When you have a negative sign in front of another negative sign, they cancel each other out. So, \(-(-5) = 5\). The two negatives become a positive.

Step 3: Compare Both Sides to Place Operand

We need to decide which operand (such as an operator like '+', '-', '*', '/') fits between \(|-5|\) and \(-(-5)\) to make a true statement. We now know both are equal to 5. Since 5 equals 5, we can place the equal sign \(=\) between the two expressions.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (e.g., -1, -2, -3). In math, these numbers express values below zero and are used in various contexts, such as temperatures or elevations below sea level. When dealing with negative numbers, it’s important to remember:

• The further left a number is on the number line, the smaller it is. This means -5 is smaller than -3.
• When subtracting a negative number, you are actually adding the positive version; for example, 5 - (-3) becomes 5 + 3.
• Multiplying or dividing two negative numbers results in a positive number.

Understanding these rules helps in effectively managing operations involving negative numbers.

Using the Number Line

The number line is a visual representation of numbers laid out in order. Every number has a specific position:

• Zero is typically at the center.
• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.

This makes the number line a useful tool for understanding concepts, like absolute value.
Absolute value, like in the exercise, represents the distance a number is from zero. Whether the number is negative or positive, this distance is always positive. On the number line, you can see how far -5 is from zero—5 steps away. This visual aid helps students grasp the concept of absolute values more intuitively.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making mathematical operations easier to perform. There are several strategies:

• Combine like terms: Group similar numbers or variables.
• Use the associative, commutative, and distributive properties to rearrange terms.
• Simplify expressions with negative numbers by carefully applying arithmetic rules.

In our step-by-step solution, simplifying \(-(-5)\) involves the rule that two negatives make a positive. So, \(-(-5) = 5\).
By recognizing patterns and applying consistent methods, simplifying expressions becomes a straightforward task, clearing the way for solving more complex problems.