Problem 30 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ 144+(-88) $$

Step-by-Step Solution

Answer

56

Step 1: Identify the Numbers

We are given two numbers: 144 and -88. The problem requires us to add these two numbers together.

Step 2: Understand the Operation

When adding a positive number and a negative number, you actually subtract the smaller number from the larger number and keep the sign of the number with the greater absolute value.

Step 3: Calculate the Absolute Difference

First, find the absolute value of each number. The absolute value of 144 is 144, and the absolute value of -88 is 88. Now subtract the smaller absolute value from the larger one: 144 - 88 = 56.

Step 4: Determine the Sign of the Result

Since the absolute value of 144 is greater than that of 88, and 144 is positive, the result of the addition will have the same sign as 144. Therefore, the result is positive.

Step 5: Write the Final Answer

Combining the result from Step 3 and the sign from Step 4, the final answer is 56.

Key Concepts

Absolute Value

The term "absolute value" refers to the distance of a number from zero on the number line, without considering its direction - positive or negative. It's like asking how far away a number is from zero, ignoring whether we're moving to the left or the right.
For instance, both 5 and -5 have an absolute value of 5 because they are both 5 units away from zero on the number line. In symbolic form, \( |5| = 5 \) and \( |-5| = 5 \).

• For positive numbers, the absolute value is the number itself.
• For negative numbers, the absolute value is the same number, but positive.

Knowing this concept helps to simplify the addition of positive and negative numbers by focusing on their absolute values rather than their signs.

Positive and Negative Numbers

Positive and negative numbers are essential in understanding integer operations, including addition. Positive numbers are those greater than zero, typically located to the right on the number line. Examples include 1, 2, 3, and so on. Negative numbers, in contrast, are less than zero and are positioned to the left on the number line, like -1, -2, -3.
When dealing with positive and negative numbers:

• Addition involving two positive numbers results in a positive sum, as you increase the distance from zero towards the right.
• Conversely, adding two negative numbers narrows you further left away from zero, leading to a negative sum.
• When adding a positive to a negative number, focus on their absolute values. The result will carry the sign of the number with the greater absolute value.

This foundational understanding aids in confidently navigating more complex operations, such as solving the problem presented.

Addition Rules

Adding integers can be tricky at first, but understanding a few simple rules can make the process much more straightforward.
Rule 1: Same Signs

• Add the numbers and keep the common sign. For example, \( 7 + 3 = 10 \) and \( -7 + (-3) = -10 \).

Rule 2: Different Signs

• Subtract the smaller absolute value from the larger one. Use the sign of the number with the larger absolute value. For example, in the operation \( 5 + (-3) \), subtract 3 from 5 to get 2. Since 5 has the larger absolute value (and is positive), the answer is 2.

Understanding these rules helps predict outcomes of integer addition efficiently, paving the way to handle such problems with ease.