Problem 130 - 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. $$ r(q-t) $$

Step-by-Step Solution

Answer

The expression \(r(q-t)\) simplifies to a positive number.

Step 1: Understand the Signs of the Variables

We are given that \(q\) is negative, \(r\) is negative, and \(t\) is positive. That means: - \(q < 0\)- \(r < 0\)- \(t > 0\).

Step 2: Determine the Sign of \(q-t\)

Since \(q\) is negative and \(t\) is positive, the expression \(q-t\) involves subtracting a positive number from a negative number. Thus, \(q-t < 0\) will be negative.

Step 3: Evaluate the Sign of \(r(q-t)\)

We have \(r < 0\) and \(q-t < 0\). The product of two negative numbers is positive. Therefore, \(r(q-t) = (-)(-) = (+)\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. We often use these in mathematics to represent value loss, debts, or directions opposite to a set standard, like below sea level or running opposite to a clock's direction.

When we have a negative number, it is symbolized by a minus sign (-) before it. For example, -5 is five units less than zero.

Understanding negative numbers is crucial in algebra because they influence the operation's outcome when added, subtracted, multiplied, or divided with other numbers, particularly when dealing with positive numbers.

It is essential to remember that when multiplied or divided by another negative number, the outcome becomes positive. The rule here is "a negative times a negative gives a positive." This rule is fundamental when simplifying algebraic expressions.

Comprehending Positive Numbers

Positive numbers are numbers greater than zero. These numbers signify quantity, height, gain, or advancement in sports scores, financial profits, or above sea level elevation.

In mathematics, they are either written with a plus sign (+) preceding them or without any sign at all. For instance, both +5 and 5 symbolize a positive five units above zero.

Positive numbers play a crucial role in arithmetic and algebra since they define positions, adjustments, and operations outcomes when combined with negative numbers.

When multiplying or dividing, the interaction between positive and negative numbers determines the result's sign, with a positive outcome only when two numbers possess the same sign.

Determining the Sign of an Expression

In algebra, sign determination is vital in understanding how different numbers in an expression interact. Understanding whether the result is positive or negative requires examining the signs involved in the operation.

When multiplying numbers, the basic rules are essential:

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

When doing operations such as adding and subtracting, pay attention to the rule of highest absolute value number determines the sign if the signs are different.

For instance, if we have an expression like \(r(q-t)\), here the operation requires understanding that multiplying \((r \times (q-t))\) needs both factors are negative, thus yielding a positive result.

Simplifying Algebraic Expressions

Algebra simplification involves reducing an expression to its simplest form. This means combining like terms, using arithmetic operations, and applying the rules of algebraic operations to make the expression as concise as possible.

Simplification is key in solving algebra problems efficiently and understanding the core problem.

For a problem like \(r(q-t)\), start by analyzing individual components and simplify them:

• Recognize the signs of each component, ensuring understanding of their positive or negative nature.
• Apply basic math operations - like multiplication - to combine and reduce the expression.
• Simultaneously apply rules concerning negative and positive numbers during each operation.

Through consistency in these steps, we ensure successful algebraic simplification.