Problem 15 - Algebra 1

Question

State the inverse operation. Multiply by \(-2\)

Step-by-Step Solution

Answer

The inverse operation is 'divide by -2'.

Step 1: Identify the effect of the original operation

The given operation is 'multiply by -2'. When a number is multiplied by -2, two things happen: first, the number doubles, and then its sign changes. For example, if we take a number 3, when it is multiplied by -2 we get -6; the number 3 has not only doubled to 6, but its sign has also changed from positive to negative.

Step 2: Find the inverse operation

The inverse operation should undo the changes caused by the original operation. So the inverse of 'multiply by -2' should divide the result by 2 and then change its sign; this operation is defined as 'divide by -2'. For instance, if we apply the operation 'divide by -2' to -6, it undoes the changes caused by 'multiply by -2', returning us to the original number 3.

Key Concepts

Multiplication and Division

Multiplication and division are two essential mathematical operations that are deeply interconnected. Multiplication involves combining equal groups, which essentially means adding a number to itself a certain number of times. For example, if you multiply 4 by 3, you add 4 together three times:

• 4 + 4 + 4 = 12,
• written as 4 × 3 = 12.

Division, on the other hand, is the process of splitting a number into equal parts or groups. It's essentially the opposite of multiplication. If you have 12 and divide it by 3, you're determining how many times 3 fits into 12:

• 12 ÷ 3 = 4.

These two operations are inverses of each other. If you multiply a number and then divide the result by the same number, you return to your original value, as long as this number isn't zero. This relationship is the foundation of many mathematical solutions, making it crucial for understanding inverse operations.

Sign Change

Sign change is an important aspect of mathematical operations, especially when dealing with positive and negative numbers. A 'sign' tells us whether a number is positive or negative. When we multiply a number by a negative value, its sign changes. A positive number becomes negative, and a negative number becomes positive.

For instance, if you multiply 5 by -1, the result is -5. Here, the positive sign of 5 changes to a negative sign. Similarly, if you multiply -8 by -1, you get 8. The negative sign changes to a positive sign. This concept is critical because it affects the final outcome of many equations and problems.

• Positive × Negative = Negative
• Negative × Negative = Positive

Understanding how sign changes work helps in solving algebraic expressions and equations, directly impacting the accuracy of your solutions.

Mathematical Operations

Mathematical operations are actions we perform on numbers to solve problems, like addition, subtraction, multiplication, and division. Each operation has its own rules and effects. For instance, addition combines quantities, while subtraction finds the difference between them.

Multiplication and division are closely related operations, as explored earlier. They can increase or decrease a quantity by a factor, and inversely undo each other, like in the case of 'multiply by -2' and 'divide by -2'.

• Understanding these operations allows for manipulating equations to solve for unknowns.
• They also enable us to understand how numbers interact with each other.

Operations that change a number's sign, like multiplying by a negative, illustrate both effect and application. They are foundational skills that help in understanding complex mathematical concepts and solve equations efficiently.