Problem 46 - Algebra 1

Question

Evaluate the expression. $$ 1.3+(-1.3)-4.2 $$

Step-by-Step Solution

Answer

-4.2

Step 1: Add 1.3 and -1.3

When we add \(1.3\) and \(-1.3\), we obtain \(0\). This is because \(-1.3\) is the additive inverse of \(1.3\), so adding these two numbers together results in zero.

Step 2: Subtract 4.2 from the Result

After obtaining \(0\) from step 1, we then subtract \(4.2\) from \(0\). Since subtracting a number is equivalent to adding its additive inverse, and the additive inverse of \(4.2\) is \(-4.2\), we add \(-4.2\) to \(0\) and get \(-4.2\).

Key Concepts

Understanding the Additive Inverse

The additive inverse of a number is a fundamental concept in mathematics. Simply put, the additive inverse is what you would add to a number to get zero. For any given number, its additive inverse is the same number with the opposite sign.
For example:

• The additive inverse of 5 is \(-5\).
• The additive inverse of \(-2\) is 2.

When you add a number to its additive inverse, they "cancel out" to zero. This idea is used frequently in simplifying expressions.The original expression in the exercise, \(1.3 + (-1.3) - 4.2\), uses the additive inverse of 1.3, which is \(-1.3\). Adding these in step 1 results in zero, showing the power of understanding additive inverses.

Subtraction as Addition of the Additive Inverse

Subtraction is often thought of as a more challenging operation than addition. However, it can be simplified by viewing subtraction as the addition of an additive inverse. When you subtract a number, it's the same as adding the opposite of that number.
For instance, when we say we're subtracting 4.2, we're really adding \(-4.2\).

• Think of subtraction as the addition of the inverse.
• This makes calculations more uniform and easier to manage.

In the exercise, after reaching zero in step 1, we subtract 4.2. Instead of complicating things, we simply add \(-4.2\) to zero, hence reaching the solution \(-4.2\). This approach helps in better understanding subtraction as a familiar addition problem.

Step-by-Step Solutions and Their Role in Learning

Step-by-step solutions are a great way to break down complex problems into more manageable parts. By understanding each piece individually, students can gain a clearer picture of the entire process.
In the given exercise:

• Step 1 shows how the additive inverse simplifies two terms to zero.
• Step 2 demonstrates subtraction by converting it to an addition problem.

This method doesn't just give you the right answer; it helps you understand why the answer is correct. When you see how each part contributes to the solution, you start to grasp underlying concepts better. Step-by-step solutions are particularly useful for practice and developing problem-solving skills in mathematics.