Problem 61 - Algebra 1: Concepts and Skills

Question

Find the sum. $$ -1+10 $$

Step-by-Step Solution

Answer

The sum of -1 and 10 is 9.

Step 1: Identify the Numbers

Identify the numbers to be added. Here the numbers are -1 and 10.

Step 2: Add the Numbers

Add the numbers together. When a positive number is added to a negative number, the result can be determined by subtracting the absolute value of the negative number from the positive number, since -1 is less than 10 in absolute value, we subtract 1 from 10.

Step 3: Write Down the Answer

After subtraction, write down the result. The result of 10 - 1 is 9.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is crucial in integer addition. Positive numbers are those greater than zero, represented without any sign or with a `+` sign. These numbers move to the right on a number line. Negative numbers, on the other hand, are less than zero and are represented with a `-` sign. They move to the left on the number line.
For example, on a number line:

• Positive numbers are like 1, 2, 3, ...
• Negative numbers are like -1, -2, -3, ...

Whenever we add a negative number to a positive number, we are effectively combining movements to the left and right on the number line. This combination results in shifting towards the direction of the number with the larger absolute value, but don’t worry, we'll discuss the absolute value in detail next!
Knowing the direction helps to easily visualize whether your result will be positive or negative. So when adding -1 and 10, we start at 10 (positive) and move 1 step left to land on 9.

Absolute Value

The absolute value of a number is the distance of that number from zero on a number line, regardless of direction. It's always a non-negative number. Think of it as the number without its sign. For example:

• The absolute value of -1 is 1.
• The absolute value of 10 is 10.

This concept helps us understand the magnitude of a number without getting confused by its sign.
When involved in addition like in our example \(-1+10\), understand that:

• The absolute value of -1 is smaller compared to 10.
• We subtract the smaller absolute value (1) from the larger one (10).

Therefore, the result is positive (9) because 10, the number with the greater absolute value, was positive. This can be useful when visualizing or calculating the sum of numbers with different signs.

Subtraction in Addition

A notable concept in integer addition is how subtraction actually appears within it. In scenarios like adding positive and negative numbers, subtraction becomes a part of our calculations.
When a positive and a negative number are combined, the operation essentially turns into a subtraction of their absolute values:

• Find the difference between their absolute values.
• The result takes the sign of the number with the larger absolute value.

In the exercise \(-1+10\):

• The absolute value of -1 is 1.
• The absolute value of 10 is 10.
• Subtract 1 from 10, which gives us 9, a positive result.

This subtraction step within addition helps clarify why the sum of a negative and a positive number often feels like subtraction: the numbers are pulling in opposite directions, the larger pull (or value) dictates the resultant direction and magnitude.