Problem 46 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-3.8+(-9.4)\)

Step-by-Step Solution

Answer

-13.2

Step 1 - Understand the Problem

The task is to add two negative numbers: -3.8 and -9.4. Remember that adding two negative numbers results in a more negative number.

Step 2 - Add the Absolute Values

Ignore the negative signs for now and add the absolute values of the numbers: 3.8 + 9.4.

Step 3 - Perform the Addition

Add 3.8 and 9.4. Starting from the rightmost digit in the tenths place, 8 + 4 = 12. Write the 2 in the tenths place and carry over the 1 to the units place. Add the units: 3 + 9 = 12, plus the carried-over 1 makes 13.

Step 4 - Reapply the Negative Sign

Since both original numbers were negative, the result is also negative. Thus, the sum of -3.8 and -9.4 is -13.2.

Step 5 - Check Using a Number Line

Optionally, verify the result using a number line. Start at -3.8 and move left 9.4 units. The endpoint should be at -13.2.

Key Concepts

absolute values

When working with addition, especially with negative numbers, understanding 'absolute values' is crucial. The absolute value of a number is its distance from zero on the number line, regardless of direction. For instance, the absolute value of both -3.8 and 3.8 is 3.8. The symbol for absolute value is two vertical bars on either side of the number, like this: \(|-3.8| = 3.8\). When adding negative numbers, we often first consider their absolute values. In the exercise, we add the absolute values of -3.8 and -9.4, which are 3.8 and 9.4 respectively. After adding these absolute values, we reapply the negative sign to the sum since both numbers are negative. This makes the final sum a negative number. Therefore, understanding absolute values helps simplify and correctly solve problems involving negative numbers.

number line

A number line is a visual representation of numbers on a straight line. It helps in understanding operations like addition and subtraction. In the earlier mentioned exercise, even though the instruction was not to use a number line, it's useful to visualize the problem.

When adding -3.8 and -9.4, you can start at -3.8 on the number line and move left (since it's negative) by 9.4 units. Each movement to the left corresponds to subtracting a positive value or adding a negative value. The more you move leftward, the more negative the result becomes.

Starting at -3.8 and moving 9.4 units to the left lands you at -13.2. This aligns with our solution and verifies that the sum of -3.8 and -9.4 is indeed -13.2. Therefore, even though you might not use a number line directly in calculations, it's a powerful tool to ensure your understanding and verify results.

negative numbers

Understanding negative numbers is essential for mastering basic arithmetic operations involving them. Negative numbers are those less than zero, and they are usually marked with a minus (-) sign. When adding two negative numbers, the general rule is that the sum becomes more negative, because both numbers pull the result further away from zero.

For example, when we add -3.8 and -9.4:

• First, we add their absolute values, getting 3.8 + 9.4 = 13.2.
• Next, we apply the negative sign back to the sum, because adding two negatives results in a negative.

Hence the sum of -3.8 and -9.4 is -13.2.

In summary, when dealing with negative numbers, always remember:

• Negative plus negative equals a more negative number.
• The process often involves adding absolute values and reapplying the negative sign.

Understanding this helps in correctly solving and interpreting the results of such arithmetic problems.