Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ -(2 x+y)[-(3 x+2 y)]=(2 x+y)(3 x+2 y) $$

Step-by-Step Solution

Answer

The real number property justifying the statement is the Inverse Property.

Step 1: Observe the expressions

We can see that the expressions on both sides of the statement seem to be related, except for the negative signs.

Step 2: Identify the significant operation

In both sides of the equation, we are multiplying two expressions which involve the same terms, i.e., $$-(2x+y)\text{ and } -(3x+2y) \text{ on the left side}$$ $$ (2x+y)\text{ and } (3x+2y) \text { on the right side }$$ Notice the presence of negative signs on the left-hand side

Step 3: Relate the expressions using the real number property

Observe that for each term being multiplied on the left side, there is a negative sign, i.e., $$[-(2 x+y)][-(3 x+2 y)]$$ We can rewrite the left-hand side of the statement as: $$[-1(2x+y)][-1(3x+2y)]$$ Now apply the Inverse Property which states that for every number a, there is a number -a, such that a + (-a) = 0. In this case, we have -1(a) = a where a is (2x+y)(3x+2y). Applying the property, we can rewrite the left side as: $$(-1 \cdot -1)[(2 x+y)(3 x+2 y)]$$ Since \((-1)(-1) = 1\), we have: $$(1)[(2 x+y)(3 x+2 y)]$$ And finally, this simplifies to the right side: $$(2 x+y)(3 x+2 y)$$

Step 4: Answer

The real number property that justifies the given statement is the Inverse Property.

Key Concepts

Real Number Properties

Real number properties are fundamental rules used in arithmetic and algebra to simplify and solve expressions. These properties include the Commutative, Associative, Distributive, Identity, and Inverse Properties, each applicable to addition and multiplication over real numbers.
One key property used in this context is the Inverse Property. The Inverse Property states that every real number has an additive inverse, which is essentially the negative of the number itself. This means if you have a number 'a', there exists another number '-a' such that their sum is zero: \( a + (-a) = 0 \).
For multiplication, the Inverse Property indicates that for any non-zero real number 'a', there is a reciprocal \( \frac{1}{a} \) allowing \( a \times \frac{1}{a} = 1 \). However, in the example provided, the focus is primarily on the property that \((-1 \times -1) = 1\), which simplifies expressions by removing negative signs from a product.

Multiplication of Expressions

When multiplying algebraic expressions, every term in one expression must be multiplied by every term in the other expression. This process is often guided by distributive property rules where you expand expressions by multiplying each term inside the parentheses.
Consider the example from the exercise: the expression \(- (2x+y)\) is multiplied by \(- (3x+2y)\). This means that:

• Each term in the first expression is scaled by the multiplication result with every term from the second expression.
• The negative signs multiply together to become positive, i.e., \((-1)\times(-1) = 1\).

This transforms the expanded product into positive terms, which significantly simplifies the algebraic expression.

Simplification of Algebraic Expressions

Simplification of algebraic expressions involves reducing them to their simplest form without changing the value of the expressions. This process can include combining like terms, using arithmetic operations, and applying properties like the Inverse Property to eliminate unnecessary complexities.
In the given problem, by applying the Inverse Property to remove negative signs, the expression \(- (2x+y)\times - (3x+2y)\) simplifies to \((2x+y)\times(3x+2y)\). Here are a few steps to help you simplify algebraic expressions:

• Identify and apply appropriate property rules like Distributive or Inverse Property.
• Eliminate redundant terms such as equal factors that cancel each other out (like the negative signs in this case).
• Rearrange and combine like terms where applicable, ensuring all possible simplifications are made while maintaining expression equivalence.

The purpose is to arrive at the most straightforward form without altering the mathematical meaning or result.