Problem 121 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(7-(-2)=5\)

Step-by-Step Solution

Answer

The statement \(7-(-2)=5\) is false. The correct statement should be \(7-(-2)=9\).

Step 1: Identify the Mathematical Operation

In the arithmetic expression \(7-(-2)=5\), a subtraction operation is written, but note that subtracting a negative number becomes addition, due to double negative rule in mathematics.

Step 2: Perform the Operation

Following the rule, perform the operation: \(7 + 2 = 9\).

Step 3: Evaluate the Statement

Compare the result with the original value on the right side of the equation, which is 5. Since \(9 \neq 5\), the initial statement \(7-(-2)=5\) is false.

Step 4: Correct the Statement

To make the equation true, replace the incorrect value (5) on the right side of the equation with the correct result (9). So, the corrected statement becomes: \(7-(-2) = 9\).

Key Concepts

Arithmetical Operations

Arithmetical operations are the fundamental building blocks of mathematics. They include addition, subtraction, multiplication, and division, which are used to calculate numerical expressions. Understanding these operations is essential in every math-related concept.

To begin with, addition involves combining numbers to get a sum. Subtraction, on the other hand, is about determining the difference between numbers, often thought of as "taking away." For instance, in the example \(7 - (-2)=5\), subtraction appears to be used, but subtracting a negative number (\(-2\)) actually turns into addition. This operation becomes \(7 + 2\), which is a common pitfall for those practicing arithmetic operations.

Multiplication and division are also crucial and can be viewed as repeated addition and subtraction respectively. The order in which these operations are performed can affect the outcome, and this is where understanding the rules and principles governing arithmetical operations is significant. Always remember to reassess an arithmetic statement or equation to ensure the operations have been executed correctly.

Negative Numbers

Negative numbers often confuse students because they behave differently from positive numbers. They are less than zero and depicted with a minus sign (\(-\)). When dealing with negative numbers, one critical rule is that "subtracting a negative" is equivalent to adding its positive counterpart.

For example, in the expression \(7 - (-2)\), the subtraction of \(-2\) becomes an addition of \(2\), resulting in \(7 + 2 = 9\). This rule is part of the broader double negative principle, which is a common hurdle in understanding negative numbers.

Here are a few basic rules for manipulating negative numbers:

• A negative number added to a positive number can either increase or decrease the absolute value, depending on which number is larger.
• Multiplying or dividing two negative numbers results in a positive product or quotient.
• Multiplying or dividing a negative number and a positive number results in a negative product or quotient.

Grasping these concepts can clarify many arithmetic operations involving negative numbers.

Equation Solving

Solving equations involves finding the value of the variable that makes the equation true. It often requires performing arithmetic operations and can include balancing both sides of an equation to maintain equality.

In the field of introductory algebra, it's essential to understand the equation solving steps. These include isolating variables, simplifying expressions, and ensuring that both sides of the equation are equivalent. For example, consider the incorrect equation \(7 - (-2) = 5\).

By understanding that \(7 - (-2)\) simplifies to \(7 + 2\), you know that the correct result should be \(9\) instead of \(5\). Thus, solving equations properly involves:

• Checking and re-evaluating calculations to correct mistakes.
• Using arithmetic rules precisely, such as those for adding and subtracting negative numbers.
• Comparing every step to ensure that the equation remains balanced and accurate.

This structured approach helps in systematically solving equations, ensuring correctness in each step.