Problem 98 - Elementary Algebra

Question

Fill in the blank with \(<,=\), or \(>\). -(-5) _____ \(-(-2)\)

Step-by-Step Solution

Answer

-(-5) > -(-2)

Step 1: Evaluate Both Sides

First, we evaluate the expressions on each side by removing the double negatives. -(-5) is evaluated as 5 because a negative sign in front of a negative number makes it positive. Similarly, -(-2) is evaluated as 2.

Step 2: Compare the Results

Now that we have simplified both expressions, we compare the results: 5 and 2. Since 5 is greater than 2, we conclude that 5 > 2.

Step 3: Fill in the Blank

Since 5 is greater than 2, the symbol '>' is the correct symbol to complete the statement. Thus: -(-5) > -(-2).

Key Concepts

Comparisons

Comparisons are fundamental in mathematics and help us understand the relationships between numbers or expressions. Essentially, they help us answer questions like "which is larger?" or "are these the same?" When comparing numbers, we often use symbols such as:

• \(<\) to show that one value is less than another.
• \(=\) to indicate that two values are equal.
• \(>\) to demonstrate that one value is greater than another.

In the given problem, we were asked to compare two expressions, -(-5) and -(-2). After evaluating them to 5 and 2 respectively, the comparison became clear: 5 is indeed greater than 2, which is why we used the greater than symbol \(>\) to fill in the blank.

Negative Numbers

Understanding negative numbers is crucial, as they have different properties compared to positive numbers. Negative numbers are those less than zero and are usually represented with a minus sign. When handling negative numbers, remember:

• A negative multiplied by a positive is negative.
• A negative multiplied by a negative is positive.

In the problem, we faced double negatives with expressions like -(-5) and -(-2). Here, removing the double negative effectively turned the values positive. That’s because two negatives cancel out, making the number opposite of negative, which is positive. Hence, -(-5) became 5, and -(-2) became 2.

Simplification

Simplification is about making expressions easier to work with, without changing their value. This involves combining like terms, reducing fractions, or in our case, handling signs. During simplification, it is essential to follow mathematical rules to avoid any mistakes.

• Combine terms with the same variables or similar structure.
• Simplify negative numbers carefully.

In this exercise, simplification involved dealing with expressions that contained double negatives. By simplifying -(-5) to 5 and -(-2) to 2, we made it easier to compare the two numbers to find the correct relationship between them. Simplifying complex expressions is a skill that will make many mathematical tasks more manageable.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a particular quantity or relationship. Understanding and manipulating expressions form the backbone of algebra and make solving equations possible. Expressions can include:

• Constant numbers, like 5 or -2.
• Variables, like \(x\) or \(y\).
• Operators, like \(+\), \(-\), \(\times\), or \(\div\).

In the exercise, the expressions were -(-5) and -(-2). Although they look complex at first, understanding how to simplify them allows you to extract meaning and perform operations, such as comparisons. Developing a good grasp of expressions will enable you to solve and understand more complex math problems efficiently.