Problem 125 - Beginning and Intermediate Algebra

Question

If \(q\) is a negative number, \(r\) is a negative number, and t is a positive number, determine whether each expression simplifies to a positive or negative number. If it is not possible to determine, state so. $$ \frac{q}{r \cdot t} $$

Step-by-Step Solution

Answer

The expression simplifies to a positive number.

Step 1: Identify Signs of Variables

Given that \(q\) is negative, \(r\) is negative, and \(t\) is positive. We have the information on the signs of each variable. \(q < 0\), \(r < 0\), and \(t > 0\).

Step 2: Simplify the Denominator

The expression in the denominator is \(r \cdot t\). Since \(r\) is negative and \(t\) is positive, the product \(r \cdot t\) results in a negative number. Thus, \(r \cdot t < 0\).

Step 3: Evaluate the Fraction

The expression is \(\frac{q}{r \cdot t}\). We have \(q < 0\) and \(r \cdot t < 0\). When dividing one negative number by another negative number, the result is positive. Therefore, \(\frac{q}{r \cdot t} > 0\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign in front of them, like \(-5\), \(-3\), or \(-0.5\). These numbers can represent a deficit, a loss, or being below a certain reference point, like sea level or temperature. Understanding negative numbers is crucial in math, as they behave differently than positive numbers.

• Adding two negative numbers results in a more negative number. For example, \(-2 + (-3) = -5\).
• Multiplying two negative numbers results in a positive number, such as \(-4 imes -2 = 8\).
• Adding a negative number to a positive number is like subtracting the absolute value of the negative number, for example, \(4 + (-3) = 1\).

When working with negative numbers, always pay attention to the signs before calculating. This helps in determining whether the result is positive or negative. In the exercise, knowing that both \(q\) and \(r\) are negative helps predict outcomes of operations with them.

Understanding Positive Numbers

Positive numbers are greater than zero and do not have any sign in front of them besides a possible plus, such as \(+5\), \(3\), or \(0.5\). They are straightforward and often used to represent quantities like height, age, or distance above ground. Recognizing positive numbers is essential to grasp algebraic equations where they interact with negative numbers.

• Adding a positive number makes a value larger. For example, \(3 + 4 = 7\).
• Multiplying a positive number with a positive or negative number has clear rules: It stays positive when multiplied by another positive, and becomes negative when interacting with a negative, such as \(3 imes -2 = -6\).
• Subtracting a positive number decreases a value, similar to adding a negative number, for example, \(7 - 2 = 5\).

Comprehending positive numbers' behavior is vital for solving algebraic problems effectively. In the exercise, understanding \(t\) as a positive number clarifies how it influences the product \(r \cdot t\), resulting in a negative outcome when combined with a negative \(r\).

Sign Determination in Algebraic Expressions

Determining the sign of an algebraic expression is fundamental in solving equations. This involves analyzing the signs of the numbers and operations given in an expression. Each arithmetic operation - addition, subtraction, multiplication, and division, affects the result's sign differently.

• For multiplication and division, an even number of negative factors results in a positive product or quotient.
• An odd number of negative factors gives a negative result. In expressions like \(\frac{q}{r \cdot t}\), identifying the signs of \(q\), \(r\), and \(t\) beforehand is a key step.
• 
  When both the numerator and denominator are negative in a fraction, the act of division turns the result positive. This is because dividing two negatives gives a positive result, as seen in fractions like \(\frac{-4}{-2} = 2\).

The exercise shows how correctly determining signs helps simplify expressions. With \(q\) and \(r\) as negative, and \(t\) positive, calculations such as \(\frac{q}{r \cdot t}\) reinforce that the outcome becomes positive.