Problem 37 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-31+(-14)\)

Step-by-Step Solution

Answer

The sum of -31 and -14 is -45.

Step 1: Understand the Problem

Identify the numbers involved and the operation required. We need to add (-31) and (-14).

Step 2: Combine the Numbers

Since both numbers are negative, add their absolute values together. The absolute values are 31 and 14. Calculate: 31 + 14 = 45.

Step 3: Apply the Sign

Because we are adding two negative numbers, the result will be negative. Therefore, -31 + (-14) = -45.

Step 4: Verify Using Number Line (Optional)

To verify, start at -31 on the number line and move 14 units to the left. You will reach -45. This confirms our solution.

Key Concepts

absolute value

Absolute value is a key concept to understand when dealing with negative numbers. It refers to the distance a number is from zero on the number line, without considering the direction. The absolute value of a number is always positive. For example:

• The absolute value of 31 is \(|31| = 31\).
• The absolute value of -14 is \(|-14| = 14\).

When adding negative numbers, we often start by finding their absolute values. This simplifies the arithmetic as we work with positive numbers.

negative numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-) in front. Adding negative numbers can seem tricky, but with practice and understanding, it gets easier. When you add two negative numbers, you essentially combine their absolute values and then apply a negative sign to the result. For instance, in the exercise \(-31 + (-14)\):

• The absolute values are 31 and 14.
• Add these absolute values: 31 + 14 = 45.
• Finally, because both original numbers were negative, our result will also be negative: \(-45\).

Remember, the rule for adding two negative numbers is to add their absolute values together and make the result negative.

verification with number line

Using a number line can be helpful to verify your addition of negative numbers. Here's how you can do it:

• Start at the first number, which is -31 in our example.
• Since you're adding -14, move 14 units to the left. Moving left means you're going in the negative direction.
• Once you finish moving 14 steps to the left, you should land on -45.

This visual representation helps to confirm that your calculation is accurate. Even though the process might seem abstract initially, using a number line can help clarify the steps and ensure your understanding is correct.