Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-6.3+(-5.2)$$

Step-by-Step Solution

Answer

The sum is \(-11.5\).

Step 1: Identify the Numbers to be Added

First, identify the numbers that need to be added together. In this case, the numbers are \(-6.3\) and \(-5.2\).

Step 2: Ignore the Negative Signs for Now

Next, the negative signs can be ignored for now in order to make the addition easier. So the problem becomes addition of \(6.3\) and \(5.2\).

Step 3: Perform the Addition

Add together the two numbers, \(6.3 + 5.2\), to get \(11.5\).

Step 4: Apply the Negative Sign

Since both original numbers were negative, the result is also negative. Hence, the sum is \(-11.5\).

Key Concepts

Negative Numbers

Understanding negative numbers is crucial when performing algebraic operations like addition. Negative numbers are simply numbers with a minus sign in front of them, for example,

• -3
• -7.5
• -0.4

They represent values less than zero and are typically found to the left of zero on the number line.
In real-life terms, you can think of negative numbers as owing money versus having money. If you have -5 dollars, it's similar to owing 5 dollars. When you add two negative numbers, as in the example you're essentially increasing the amount you owe. For instance, if you owe 6.3 (represented by -6.3) and you owe an additional 5.2 (represented by -5.2), then you owe a total of 11.5, which is written as -11.5.

Addition of Decimals

Addition of decimals can sometimes be tricky, but with careful alignment, you can master it. Decimals are numbers with a point that divides the whole number from the fractional part. For instance, in 6.3, the 6 is a whole number, while the .3 is the fractional part. To add decimals like 6.3 and 5.2:

• Align the numbers by the decimal point, so that each digit is in the correct column (e.g., ones, tenths).
• Add each column starting from the right, just like you would in regular addition.
• Keep the decimal point in the same position in the result.

This alignment ensures that all digits are added accurately, resulting in an accurate sum of 11.5 in our exercise. This process is exactly the same, regardless of whether the numbers are positive or negative—just be sure to handle the signs appropriately at the end.

Signed Numbers

Signed numbers are a fundamental concept in algebra, representing both positive and negative values. The sign of a number indicates its direction on the number line:

• Positive numbers (without a visible sign or explicitly marked with a "+") indicate a value greater than zero.
• Negative numbers (marked with a "-") indicate a value less than zero.

Handling signed numbers in addition requires understanding how their signs impact the operation: - **Adding Two Negative Numbers**: Just like in our exercise, when both numbers are negative, the sum is negative. Simply add the absolute values and use the negative sign for the result.
- **Adding Two Positive Numbers**: If both numbers are positive, the sum is also positive. Simply perform regular addition.
- **Adding a Positive and a Negative Number**: Determine the larger absolute value to find the sign of the result. Use subtraction of the smaller absolute value from the larger. If the positive number is larger, the result is positive. Otherwise, it is negative.