Product Less Than Zero: Find The Negative Solution!

Let's explore how to determine which expression results in a product less than zero. This involves understanding the rules of multiplying positive and negative numbers. A negative product indicates that the result of the multiplication is a negative number. We'll go through each option step by step.

Understanding the Basics of Multiplication with Negative Numbers

Before diving into the options, let's quickly recap the rules of multiplication with negative numbers:

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

Essentially, if you have an odd number of negative signs in your multiplication, the result will be negative. If you have an even number of negative signs (or none at all), the result will be positive. With these rules, we can tackle the given expressions and identify the one that yields a product less than zero.

To make it easier to understand, we'll illustrate each rule with examples:

1. Positive × Positive = Positive: For example, 2 × 3 = 6. Both numbers are positive, and the result is a positive number.
2. Negative × Negative = Positive: For example, (-2) × (-3) = 6. Both numbers are negative, but since there are two negative signs, the result is a positive number.
3. Positive × Negative = Negative: For example, 2 × (-3) = -6. One number is positive, and the other is negative, resulting in a negative number.
4. Negative × Positive = Negative: For example, (-2) × 3 = -6. One number is negative, and the other is positive, resulting in a negative number.

These rules are fundamental in determining the sign of the product, and understanding them thoroughly will help in solving more complex problems involving multiplication of signed numbers. Now, let's apply these rules to each of the given options.

Evaluating Each Option

Now, let's evaluate each option to determine which one has a product less than zero (i.e., a negative product).

(a) $(-4.7)(0)$

In this option, we are multiplying -4.7 by 0. Any number multiplied by zero is zero. Therefore:

$(-4.7)(0) = 0$

Since 0 is neither positive nor negative, this option does not result in a product less than zero.

(b) $(-0.6)(-1.7)$

Here, we are multiplying two negative numbers: -0.6 and -1.7. According to our rules, the product of two negative numbers is positive:

$(-0.6)(-1.7) = 1.02$

The result is 1.02, which is a positive number. Therefore, this option does not have a product less than zero.

(c) $(5.9)(-2.6)(13)$

In this option, we have three numbers being multiplied: 5.9, -2.6, and 13. We have one negative number (-2.6) and two positive numbers (5.9 and 13). The product will be:

$(5.9)(-2.6)(13) = (5.9 \times 13) \times (-2.6) = 76.7 \times (-2.6) = -199.42$

The result is -199.42, which is a negative number. Therefore, this option has a product less than zero.

(d) $(-9.3)(1.7)(-4)$

In this option, we are multiplying three numbers: -9.3, 1.7, and -4. We have two negative numbers (-9.3 and -4) and one positive number (1.7). The product will be:

$(-9.3)(1.7)(-4) = (-9.3 \times -4) \times (1.7) = 37.2 \times 1.7 = 63.24$

The result is 63.24, which is a positive number. Therefore, this option does not have a product less than zero.

Conclusion

After evaluating each option, we can conclude that only option (c) results in a product that is less than zero.

• (a) $(-4.7)(0) = 0$ (Not less than zero)
• (b) $(-0.6)(-1.7) = 1.02$ (Not less than zero)
• (c) $(5.9)(-2.6)(13) = -199.42$ (Less than zero)
• (d) $(-9.3)(1.7)(-4) = 63.24$ (Not less than zero)

Therefore, the correct answer is:

(C) $(5.9)(-2.6)(13)$

This expression has a product that is less than zero.

In summary, to determine whether a product is less than zero, count the number of negative factors. If there is an odd number of negative factors, the product will be negative. If there is an even number of negative factors (or none), the product will be positive. Understanding these rules is key to quickly identifying the correct answer.

When dealing with multiplication involving both positive and negative numbers, it's important to keep track of the signs. Remember that multiplying two numbers with the same sign (both positive or both negative) yields a positive result, while multiplying two numbers with different signs yields a negative result. This principle extends to multiple factors: an odd number of negative factors results in a negative product, while an even number results in a positive product.

For example, if we have the expression $(-1) \times (-1) \times (-1)$, there are three negative factors, so the product is negative: $(-1) \times (-1) \times (-1) = -1$. On the other hand, if we have the expression $(-1) \times (-1) \times (-1) \times (-1)$, there are four negative factors, so the product is positive: $(-1) \times (-1) \times (-1) \times (-1) = 1$.

Understanding these basic rules can help you quickly and accurately determine the sign of a product without having to perform the full multiplication.

For additional resources and further learning on mathematical principles, you might find helpful information on websites like Khan Academy's Arithmetic Pre-algebra section.