Problem 40 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ -32.6-(-9.8) $$

Step-by-Step Solution

Answer

The result is \(-22.8\).

Step 1: Understanding the Problem

The exercise requires us to calculate the expression \(-32.6 - (-9.8)\). We see that there is a negative sign before the parenthesis which contains another negative number. This is a subtraction of a negative number from another negative number.

Step 2: Simplifying the Expression

Subtracting a negative number is equivalent to adding the positive equivalent of that number. Therefore, \(-32.6 - (-9.8)\) becomes \(-32.6 + 9.8\).

Step 3: Performing the Addition

Now we need to add \(-32.6\) and \(9.8\). We do this by direct arithmetic: 1. Keep the sign of the number with the greater absolute value, which is \(-32.6\) in this case.2. Subtract the smaller absolute value from the larger one: \(32.6 - 9.8 = 22.8\).3. Attach the sign from the larger absolute value, giving us the result \(-22.8\).

Key Concepts

Subtraction of Real Numbers

Subtraction of real numbers can sometimes seem tricky, but it's mostly about understanding the relationship between the numbers involved. When you see an equation like

• \( -32.6 - (-9.8) \)

we are working with a subtraction operation that involves two negative numbers. The crucial part to remember is that subtracting a negative number is the same as adding its positive counterpart. So, when you subtract

• \(- (-9.8)\)

it becomes

• \(+ 9.8\).

This simplifies the expression and changes the operation from subtraction to addition. Recognizing and using this rule can help simplify many arithmetic problems.Key rule of thumb for subtraction:

• Subtracting a negative is the same as adding a positive.
• Always transform the expression to simplify the calculations.

Addition of Real Numbers

Adding real numbers involves straightforward arithmetic once you know the rules about signs. After simplifying the subtraction from the previous step, we have

• \( -32.6 + 9.8 \).

When adding a positive number to a negative number, the steps are:First, identify the number with the larger absolute value. In this case, it's

• \( 32.6 \) (ignoring the negative sign for determining the size).

The sign of the result is determined by this larger absolute value. Here, it's negative.Next, subtract the smaller absolute value from the larger one:

• \( 32.6 - 9.8 = 22.8 \).

Then attach the appropriate sign: since

• \( 32.6 \) was negative, the result is
• \( -22.8 \).

These operations combine to simplify real number addition and reduce mistakes.

Negative Numbers in Arithmetic

Understanding negative numbers is crucial in arithmetic as they often appear in various math problems. Negative numbers are simply numbers less than zero, and they can be challenging when used in operations like subtraction and addition.A few things to keep in mind when working with negative numbers:

• Adding a negative number is the same as subtraction.
• Subtracting a negative number is equivalent to adding the positive version of that number.
• Always pay attention to the signs, as they determine the direction of the operation on the number line.

For example,

• \(-32.6\) is positioned 32.6 units left of zero on the number line.
• When \( -32.6 \) is combined with \( (+9.8) \), we effectively move 9.8 units to the right from \(-32.6\).

By understanding these properties and visualizing operations on a number line, calculations involving negative numbers become clearer and less prone to errors.