Problem 7 - Contemporary Precalculus

Question

\(b, c,\) and \(d\) are real numbers such that \(b<0\) \(c>0,\) and \(d<0 .\) Determine whether the given number is positive or negative. $$b c-b d$$

Step-by-Step Solution

Answer

Answer: Negative

Step 1: Find the sign of \(bc\)

Since \(b < 0\) and \(c > 0\), their product \(bc\) will be negative (it is a negative number multiplied by a positive number). So, \(bc < 0\).

Step 2: Find the sign of \(-bd\)

Since \(b < 0\) and \(d < 0\), their product \(bd\) will be positive (it is a negative number multiplied by a negative number). Thus, \(bd > 0\). Now, let's consider the negative of \(bd\): \(-bd < 0\) (we are negating the positive product).

Step 3: Determine the sign of \(bc - bd\)

From Steps 1 and 2, we know that \(bc < 0\) and \(-bd < 0\). Therefore, when we add the two negative numbers together, the sum will also be negative. So, the expression $$bc - bd < 0$$ is negative.

Key Concepts

Multiplication of Negative Numbers

Understanding the multiplication of negative numbers is essential in the realm of real numbers. When multiplying two negative numbers, the product is always positive. This might seem counterintuitive at first, but it aligns with the rules of arithmetic. For example:

• When you multiply the negative number -2 by another negative number -3, the result is a positive 6.
• This holds because multiplying by a negative reverses the sign of the other number, and reversing twice returns to a positive sign.

So, when dealing with expressions involving negative numbers, the context of multiplication determines the sign of the product. The exercise illustrates this when calculating the product of two negative numbers, like \(bd\), resulting in a positive product.

Negation of a Product

Negating a product is simply reversing its sign. If you have a positive product and negate it, the result becomes negative, and vice versa. This is crucial when determining the sign in expressions with multiple terms. For example, if you have a positive product like \(bd > 0\), negating it yields \(-bd < 0\). This process is straightforward:

• Take the product of the numbers as it stands.
• Multiply by -1, effectively reversing the sign.

This concept is used in the exercise when taking \(bd\), which is positive, and negating it to \(-bd\), making it a negative term in the expression \(bc - bd\). Thus, understanding negation is vital to correctly interpret the direction of inequalities when subtracting terms.

Properties of Inequalities

Inequalities function like an extension of the basics of arithmetic, adding a layer of comparison. Understanding their properties helps in making correct conclusions about expressions. Consider these fundamental properties:

• Adding or subtracting the same number from both sides of an inequality does not change the inequality's direction.
• Multiplying or dividing both sides by a positive number maintains the inequality's direction.
• However, multiplying or dividing by a negative number reverses the inequality's direction.

In our exercise, we use these properties to deal with expressions like \(bc - bd\). Knowing \(bc < 0\) and \(-bd < 0\) allows us to say confidently that their sum is negative. The properties of inequalities guide us to understand why two negative quantities result in a further negative value when combined. This clarity is crucial for confidently interpreting and solving such mathematical problems.