Problem 6 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$(-1)(48)=-48$$

Step-by-Step Solution

Answer

Multiplicative property of -1.

Step 1: Identify Operations and Terms

Observe the equation \((-1)(48) = -48\). This equation involves multiplication of two numbers: \(-1\) and \(48\). The result is \(-48\).

Step 2: Understand the Property Concept

The statement involves the multiplication of a negative number with a positive number. The negative sign is kept with the product. Essentially, multiplying a number by -1 yields its additive inverse.

Step 3: State the Mathematical Property

The property that explains this equation is the "Multiplicative Property of -1". This property states that for any real number \(a\), multiplying by \(-1\) results in the negative of that number, which is \(-a\). In this case, multiplying \(48\) by \(-1\) gives \(-48\).

Key Concepts

Multiplicative Property of -1

The Multiplicative Property of -1 is an essential algebraic property stating that any real number multiplied by -1 results in its opposite, also known as its negative. This property is particularly useful when dealing with negative numbers and helps us understand how signs change in multiplication.
Let's break it down with an example: when you multiply 48 by -1, as in the equation \((-1)(48) = -48\), the result is -48. This is because the operation flips the number's sign. Here are key points to remember:

• Multiplying a positive number by -1 results in a negative number.
• If the number is already negative, multiplying by -1 will make it positive.
• This property doesn't change the magnitude of the number, only its sign.

In summary, multiplying any number by -1 effectively "inverts" it around zero.

Commutative Property of Addition

The Commutative Property of Addition is another fundamental idea in algebra. It states that numbers can be added in any order, and the sum will remain the same. This property is famously demonstrated by the equation \(3 + (-4) = (-4) + 3\).
This means that regardless of the sequence in which the numbers appear, the result will not change. Understanding this can simplify computations and solve problems more flexibly. Here are the essential aspects of this property:

• Order of addition does not affect the sum.
• Mathematically, if you have two numbers \(a\) and \(b\), then \(a + b = b + a\).
• This property is extremely useful for mental math and rearranging terms to make addition easier.

By applying this property, adding numbers becomes more intuitive, especially when dealing with larger sets of numbers or variables.

Additive Inverse Concept

The Additive Inverse Concept is a vital part of understanding how numbers interact in arithmetic. The additive inverse of a number is the value that, when added to the original number, results in zero. This is closely tied to the idea of getting back to the origin or balance point on a number line.
For instance, the additive inverse of 5 is -5 because \(5 + (-5) = 0\). Here's what you should know:

• Every real number \(a\) has an additive inverse, which is \(-a\).
• When the two are added together, the result is always zero.
• The concept helps in simplifying equations by "eliminating" terms through addition.

Understanding the additive inverse is crucial when solving algebraic equations, as it allows us to isolate variables and work toward the solution more efficiently.