Problem 31 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-121+170$$

Step-by-Step Solution

Answer

The result is 49.

Step 1: Identify the Numbers and Their Signs

We are given two numbers to add: -121 and +170. The negative sign before 121 indicates that 121 is a negative number, while the positive number 170 has a positive sign implicitly given.

Step 2: Apply the Rule for Adding Numbers with Different Signs

When combining a negative number with a positive number, subtract the smaller absolute value from the larger absolute value. Here, compare 121 (the absolute value of -121) and 170.

Step 3: Subtract the Smaller Absolute Value from the Larger

Subtract 121 from 170: \[ 170 - 121 = 49 \]

Step 4: Determine the Sign of the Result

Since the positive number (170) has a greater absolute value than the negative number (121), the result will take the sign of the positive number. Therefore, the result is positive.

Key Concepts

Absolute Value

Understanding absolute value is key to grasping integer operations. An absolute value represents the distance a number is from zero on the number line. It is always a non-negative number.

For any integer, whether negative or positive, the absolute value is the number itself without its sign. This means:

• The absolute value of \(-121\) is \(121\).
• The absolute value of \(170\) is \(170\).

Absolute value helps standardize numbers for easier comparison and operation, such as addition or subtraction of integers.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign \((-\)). They can be understood as numbers that are in the opposite direction on the number line from positive numbers.

Important points about negative numbers include:

• They signify a decrease, loss, or absence of quantity.
• A negative number has a lower position than zero on the number line.
• The larger the absolute value of a negative number, the smaller the number itself (for example, \(-121\) is less than \(-50\)).

Negative numbers are vital in various contexts such as finance (debt) or temperature (below zero). Adding them to positive numbers narrows the gap, leading to a balanced result.

Positive Numbers

Positive numbers are all numbers greater than zero. They are written without a sign or sometimes with a plus sign \((+\)).

Here's what you need to know about positive numbers:

• They represent gain, increase, or surplus of quantity.
• They are located to the right of zero on the number line.
• Adding positive numbers moves you further away from zero, increasing the overall value.

Positive numbers are used in everyday life: whether counting objects, measuring degrees, or handling gains in financial scenarios.

Addition Rules

Understanding the rules of addition, especially with integers, simplifies solving problems like \(-121 + 170\). The fundamental rule for adding numbers with different signs is:

• Find the absolute values of both numbers involved.
• Subtract the smaller absolute value from the larger absolute value.
• The result takes the sign of the number with the larger absolute value.

When adding integers:

• If both numbers are positive, add them and the result is positive.
• If both numbers are negative, add the absolute values and then apply a negative sign to the result.
• If one number is positive and the other is negative, follow the above steps as shown.

These rules ensure that no matter the signs, you can always find the correct sum of integers.