Problem 10 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -7+(-4) $$

Step-by-Step Solution

Answer

The sum is -11.

Step 1: Identify the Numbers

The problem asks for the sum of two numbers, \(-7\) and \(-4\). Note that both numbers are negative.

Step 2: Add Absolute Values

Disregard the negative signs and add the absolute values of the numbers.Calculate \(7 + 4 = 11\).

Step 3: Apply Negative Sign to the Sum

Since both numbers are negative, the sum will also be negative. Apply a negative sign to the sum obtained from Step 2.Thus, \(-11\).

Key Concepts

Absolute Value

Absolute value denotes the distance a number is from zero on the number line, regardless of direction. It transforms any number into a positive or zero value, essentially ignoring any negative signs. For instance, the absolute value of \(-7\) is 7, and for \(-4\), it is 4.
Absolute value is crucial when dealing with negative numbers because it allows us to treat them as if they're positive when adding or subtracting. By focusing on the magnitude of the number, rather than its sign, calculations become more straightforward.
This concept is incredibly helpful in mathematics, especially in operations involving negative integers as it simplifies addition, welcoming the correct application of signs in the final steps.

Integers

Integers are a fundamental component of mathematics and include all positive and negative whole numbers, including zero. They form an infinite sequence: ..., -3, -2, -1, 0, 1, 2, 3, and so on.
Integers are crucial in various math operations, including addition, subtraction, multiplication, and division. They are used to represent quantities without fractions, which simplifies calculations and is essential to understanding more complex mathematical concepts.
When dealing with integers:

• Positive integers are greater than zero.
• Negative integers are less than zero.
• Zero is neutral and is neither positive nor negative.

Understanding integers is key to mastering number operations, especially when they involve negative values.

Addition of Integers

Adding integers is straightforward if you grasp the rules associated with their signs. When both integers share the same sign, add their absolute values and give the sum that common sign.
Consider the example from the exercise: adding \(-7\) and \(-4\). Start by calculating the absolute values, which are 7 and 4 respectively, yielding a sum of 11. Since both integers are negative, the result will also be negative. Therefore, \(-7 + (-4)\) becomes \(-11\).
Here are some key points to remember with integer addition:

• Add the absolute values if the integers have the same sign, then use that sign in the result.
• If integers have different signs, subtract the smaller absolute value from the larger one, and use the sign of the number with the larger absolute value.

Understanding these rules will help you tackle more complex mathematical problems where adding integers is required.