Problem 10 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -7+(-4) $$

Step-by-Step Solution

Answer

The sum of -7 and -4 is -11.

Step 1: Identify the Terms

In the expression \(-7 + (-4)\) the terms are \(-7\) and \(-4\). Both are negative numbers.

Step 2: Understand Adding Negative Numbers

When adding two negative numbers, you find the sum of their absolute values and then apply a negative sign to the result. The absolute values of the terms are: \(|-7| = 7\)and\(|-4| = 4\).

Step 3: Find the Sum of Absolute Values

Add the absolute values of 7 and 4 together. Calculate \(7 + 4 = 11\). So, the sum of the absolute values is 11.

Step 4: Apply the Negative Sign to the Result

Since both numbers are negative, the result must also be negative. Therefore, add a negative sign to the sum of absolute values: \(-11\).

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on a number line, without considering its direction. This means that both positive and negative numbers have the same absolute value if they are equidistant from zero. For example, both \(5\) and \(-5\) have an absolute value of 5.
Understanding absolute value is crucial because it simplifies operations like adding or subtracting negative numbers. It allows us to focus solely on the size of the number, ignoring whether it is positive or negative.

• The absolute value of any positive number is the number itself.
• The absolute value of zero is 0.
• The absolute value of a negative number is its positive form.

In the exercise \(-7 + (-4)\), the absolute values are \(|-7| = 7\) and \(|-4| = 4\). We use these absolute values to further simplify and find the sum of negative numbers.

Negative Numbers

Negative numbers are those less than zero and typically represented with a minus sign. They exist to indicate loss, deficiency, or a position below a baseline, such as below sea level or colder temperatures on a thermometer.
In mathematics, understanding the properties of negative numbers is fundamental, especially when it comes to arithmetic operations, like addition and subtraction.

• Negative numbers decrease in value as they approach more negative values.
• Addition with negative numbers might initially seem challenging due to its counter-intuitive nature.
• When adding two negative numbers, you actually sum their absolute values.

In the problem \(-7 + (-4)\), both numbers are negative, leading to a more negative result as we add their magnitudes.

Integer Addition

Adding integers involves combining whole numbers, which can be positive or negative. Integer addition follows a basic principle: - If both integers have the same sign (both positive or both negative), add their absolute values and keep the sign.
- If they have different signs, subtract the smaller absolute value from the larger and use the sign of the integer with the larger absolute value.

• When adding positive integers, simply add and retain the positive sign.
• Negative integers require adding absolute values, then affixing a negative sign.
• For mixed signs, calculate the difference and apply the sign of the larger absolute value.

For instance, in \(-7 + (-4)\), both integers are negative. Thus, add their absolute values to get 11, then assign a negative sign to get a total of \(-11\). This consistent application of rules helps in accurately computing any integer addition.