Problem 18 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(12+(-22)\)

Step-by-Step Solution

Answer

-10

Step 1: Understand the Signs

Identify the signs of the numbers. Here, 12 is positive and -22 is negative.

Step 2: Subtract the Smaller Absolute Value from the Larger Absolute Value

Since the numbers have different signs, subtract the smaller absolute value (12) from the larger absolute value (22). 22 - 12 = 10

Step 3: Assign the Correct Sign

The sign of the result comes from the number with the larger absolute value. Since -22 has the larger absolute value and is negative, the result is -10.

Step 4: Combine the Results

The final result is -10.

Step 5: Check with Number Line (if needed)

Optionally, use a number line to verify: Starting at 12, move 22 units to the left, which would give -10.

Key Concepts

Positive and Negative Numbers

When dealing with integer addition, understanding positive and negative numbers is crucial.
A positive number is any number greater than zero, such as 1, 2, and 12.
Positive numbers are written without a sign or with a '+' sign.
On the other hand, negative numbers are those less than zero, such as -1, -2, and -22, and they are always written with a '-' sign.

When adding numbers of different signs, the process involves subtracting the smaller absolute value from the larger absolute value.

• Example: For 12 + (-22), subtract 12 from 22. Then, assign the sign of the number with the larger absolute value to the result.

It helps to remember that:

• Adding two positive numbers always gives a positive result.
• Adding two negative numbers always gives a negative result.
• Adding a positive number to a negative number (or vice versa) may give a positive or negative result, depending on the absolute values.

Absolute Value

Another important concept in integer addition is the absolute value of a number.
The absolute value of a number is its distance from zero on the number line, regardless of direction.
It is always a non-negative number.

For example:

• The absolute value of 12 is 12 (written as \(|12| = 12\)).
• The absolute value of -22 is 22 (written as \(|-22| = 22\)).

Absolute values help simplify the process of adding integers with different signs.
By converting negative numbers to their absolute values, you can easily perform subtraction to find the magnitude of the result.
Remember: After performing the operation, the sign of the result is determined by the number with the larger absolute value.
If the number with the larger absolute value is negative, the final result will be negative.
If it is positive, the result will be positive.

Number Line

A number line is a visual tool that can help understand the addition of integers.
It is a straight line with numbers placed at equal intervals along its length, starting with zero in the middle.
Positive numbers are placed to the right of zero, while negative numbers are to the left.

To add integers using a number line:

• Start at the position of the first number.
• Move right if adding a positive number or move left if adding a negative number.

For example, to verify \(12 + (-22)\), start at 12:

• Since you are adding -22, move 22 units to the left.
• You will land on -10, confirming that 12 + (-22) = -10.

The number line is beneficial for visual learners and can help confirm your calculations.
Even though you may not always use it to solve problems directly, it is an excellent tool for checking your work and understanding the relationship between numbers.