Problem 129 - 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. $$ t(q+r) $$

Step-by-Step Solution

Answer

The expression simplifies to a negative number.

Step 1: Identify the Given Information

We are given that \( q \) is a negative number, \( r \) is a negative number, and \( t \) is a positive number. We need to consider the expression \( t(q + r) \).

Step 2: Evaluate the Sum Inside the Parentheses

Since \( q \) and \( r \) are both negative numbers, their sum \( (q + r) \) will also be negative. This is because adding two negative numbers results in a negative number.

Step 3: Simplify the Expression

The expression \( t(q+r) \) can be rewritten as \( t \times (q + r) \). We already determined that \( q + r \) is negative, and \( t \) is positive, so we need to multiply a positive number by a negative number.

Step 4: Determine the Sign of the Product

Multiplying a positive number (\( t \)) by a negative number (\( q + r \)) results in a negative number. Therefore, the product \( t(q+r) \) is negative.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (−) in front. For example, −3 is a negative number. Negative numbers are used to denote values below a starting point or, for example, temperatures below freezing or depths below sea level.
It's essential to remember some key points about negative numbers:

• Any negative number is always less than any positive number.
• When you add a negative number to another negative number, the result is a more negative number. For example, \(-2 + (-3) = -5\).
• Subtracting negative numbers is the same as adding their positive equivalent—for example, \(-5 - (-2) = -5 + 2 = -3\).

Understanding these concepts is crucial when dealing with algebraic expressions involving negative numbers.

Exploring Positive Numbers

Positive numbers are numbers greater than zero, and they do not have any sign in front because they're naturally understood to be positive. They appear as counting numbers like 1, 2, 3, etc. Positive numbers are integral in representing values above a certain point or profit in financial calculations. Key aspects to note are:

• Positive numbers are greater than zero.
• Adding two positive numbers gives a positive result.
• Subtracting a smaller positive number from a larger positive number also results in a positive number.
• Positive numbers are always placed to the right of zero on a number line.

This fundamental understanding helps when multiplying, adding, or subtracting positive and negative numbers in algebraic equations.
Recognizing when numbers are positive is crucial for simplifying algebraic expressions correctly.

Multiplication of Integers

When multiplying integers, understanding the rules surrounding signs is vital. The multiplication of integers involves straightforward rules that determine the sign of the result:

• Multiplying a positive number by a positive number gives a positive product, e.g., \( 2 \times 3 = 6 \).
• Multiplying a negative number by a positive number results in a negative product, e.g., \( -4 \times 2 = -8 \).
• Multiplying two negative numbers results in a positive product, e.g., \( -3 \times -2 = 6 \).
• If the number of negative numbers in the multiplication is odd, the result is negative.

For the given expression \(t(q + r)\), understanding that we multiply a positive number \(t\) with the negative sum \(q + r\) results in a negative product is critical. Using these rules helps in predicting and simplifying such expressions correctly and efficiently.