Problem 12 - Algebra A Combined Function

Question

Subtract. \(-8-4\)

Step-by-Step Solution

Answer

The result is -12.

Step 1: Understand the Problem

The problem asks us to subtract 4 from -8, which can also be understood as adding the negative of 4 to -8. The operation is: \(-8 - 4\).

Step 2: Rewrite the Subtraction as Addition

Subtraction can be converted to addition by adding the negative. Rewrite the problem as \(-8 + (-4)\). This makes it clearer that we are adding two negative numbers.

Step 3: Add the Negative Numbers

When adding -8 and -4, you combine their absolute values and retain the negative sign. \(-8 + (-4) = -(8 + 4) = -12\).

Step 4: Verify the Solution

Check the answer by considering what happens on a number line. Starting at -8, moving 4 units in the negative direction leads to -12.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign \(-\) in front of them. Negative numbers are used to describe quantities that are below a defined point, like zero, or to signify a loss or deficiency. For example, -8 represents a position 8 units to the left of zero on the number line.

• Temperatures below zero are negative numbers.
• Debt or owing money can be represented as negative numbers.
• In gaming, losing points can be represented with negative numbers.

Understanding negative numbers is crucial as they are often encountered in real life. They form an essential part of arithmetic, especially when dealing with subtraction and addition.

Addition as Inverse of Subtraction

Subtraction can be thought of as adding the opposite. It can often be more intuitive to replace subtraction with addition of a negative. Noticing this relationship helps simplify math problems. For instance, the problem \(-8 - 4\) can be rewritten as \(-8 + (-4)\). Here, you are adding the negative of 4.

• This conversion is especially useful in simplifying operations with negative numbers.
• Recognizing this inverse relationship makes it easier to use mental math.

Remember, turning subtraction into addition is an essential strategy that also provides a clearer insight into how negative numbers interact.

Number Line

A number line is a helpful tool to visualize numbers. It is a straight line with numbers placed at equal intervals along its length. Zero is typically at the center, with positive numbers to the right and negative numbers to the left. Using a number line can make understanding arithmetic operations, like addition and subtraction, straightforward:

• For negative numbers, you move left from zero.
• Subtracting a positive number means moving further left on the number line.
• Addition can be visualized as moving to the right, while subtraction moves left.

In our example, starting at -8 and moving 4 units left lands us at -12. Visually exploring through a number line helps solidify abstract concepts.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, disregarding its sign. In simpler terms, it's the non-negative value of a number.For example:

• The absolute value of -8 is 8.
• The absolute value of 4 is 4.

Understanding absolute values helps in combining negative numbers. In \(-8 + (-4)\), you use the absolute values of 8 and 4 to find the result. By adding \((8 + 4)\), you get 12, and since the original numbers were negative, the result is negative. Hence, \(-12\) is the final answer. Knowing the concept of absolute value is crucial for simplifying calculations and understanding negative number behaviour.